Hungerford Algebra 第 II-IV 章定义、定理与性质摘录

说明：本笔记按原书正文顺序整理第 II、III、IV 章的编号 Definition / Theorem / Proposition / Lemma / Corollary 条目；Proposition 按“命题/性质”处理。以下为 PDF 文本层抽取的无证明摘录，公式和特殊符号可能保留原文本层的识别形态，必要时请对照原 PDF。

抽取统计

• 编号条目总数：240
• 第 II 章：71 条
• 第 III 章：82 条
• 第 IV 章：87 条
• 类型统计：定理 113；命题/性质 22；引理 27；推论 40；定义 38

目录

• [[#Chapter II: The Structure of Groups|Chapter II: The Structure of Groups]]
• [[#Chapter III: Rings|Chapter III: Rings]]
• [[#Chapter IV: Modules|Chapter IV: Modules]]
• [[#未编号术语定义候选|未编号术语定义候选]]

Chapter II: The Structure of Groups

II.1 Free Abelian Groups

1. Theorem 1.1（定理）

Theorem 1.1. The following conditions on an abelian group F are equivalent.
 (i) F has a nonempty basis.
 (ii) F is the (internal) direct sum of a family of infinite cyclic �-ubgroups.
 (iii) F is (isomorphic to) a direct sum ofcopies ofrhe additive group Z of integers.
 (iv) There exists a nonempty set X and a function L : X � F with the following
property: given an abelian group G and function f : X � G, there exists a unique homo­
morphism ofgroups f : F -+ G such that ft = f. In other words, F is a free object in the
category of abelian groups.

 An abelian group F that satisfies the conditions of Theorem 1 . 1 is called a free
abelian group (on the set X). By definition the trivial group 0 is the free abelian group
on the null set 0 .

2. Theorem 1.2（定理）

Theorem 1.2. Any two bases of a free abelian group F have the same cardinality.

 The cardinal number of any basis X of the free abelian group F is thus an invari­
ant of F; IX/ is called the rank of F.

3. Proposition 1.3（命题/性质）

Pro position 1.3. Let F. be the free abelian group on the set X . and F2 the free abelian
group on the 5et X 2 . Then F. "" F2 ifand only ifFt and F2 have the same rank (that is,
IX./ = IX 2/ ).

 REMARK . Proposition 1 .3 is also true for arbitrary nonabelian free groups (as
in Section 1.9); see Exercise 1 2 .

4. Theorem 1.4（定理）

Theorem 1.4. Every abelian group G is the homomorphic image of a free abelian
group of rank !X I, where X is a set of generators of G.

5. Lemma 1.5（引理）

Lem ma 1.5. If{ x. , . . . , X n } is a basis ofa free abelian group F and a € Z, then for all
i � j { x�, . . . , Xj- t ,Xj + axi,Xj+I, . . . , Xn } is also a basis ofF.

6. Theorem 1.6（定理）

Theorem 1.6. If F is a free abelian group offinite rank n andG is a nonzero subgroup
of F, then there exists a basis { x., . . . , Xn } of F, an integer r ( 1 < r < n) and positive
integers d., . . . , dr such that d1 I d 2 I · · · I d r and G is free abelian with basis
 { d1x 1, . . . , drXr J .

 REMARKS. Every subgroup of a free abelian group of (possibly infinite) rank a
is free of rank at most a; see Theorem IV.6.1 . The notation "d1 I d2 l . . I d," means
 .




... d1 divides d2 , d2 divides da., etc."

7. Corollary 1.7（推论）

Corollary 1. 7 . JfG is a finitely generated abelian group generated by n elements, then
every subgroup H of G may be generated by m e/e1nents with m < n.

 The corollary is false if the word abelian is omitted (Exercise 8).

II.2 Finitely Generated Abelian Groups

8. Theorem 2.1（定理）

Theorem 2.1. Every finitely generated abelian group G is (isomorphic to) a finite
direct sum ofcyclic groups in which the finite cyclic summands {if any) are of orders
m 1 , . . , fit, where m1 > 1 and m1 I m2 l · · · lmt .
 .

9. Theorem 2.2（定理）

Theorem 2.2. Every finitely generated abelian group G is (isomorphic to) a finite
direct sun1 ofcyclic groups, each of which is eirher infinite or oforder a power ofaprime.

10. Lemma 2.3（引理）

Lem ma 2.3. If m is a positive integer and m = Ptn1P2n2 . Ptnt (p h . . . , Pt distinct
 · ·




primes and each ni > 0), then Zm f"V Zp1nl EB ZP2n� ffi · · · EB ZPtn t .

11. Corollary 2.4（推论）

Corollary 2.4. JfG is a finite abelian group oforder n, then G has a subgroup oforder
m for every positive integer m that divides n.
 k

12. Lemma 2.5（引理）

Lemma 2.5. Let G be an abelian group, m an integer and p a prime integer. Then each
of the following is a subgroup ofG:
 (i) mG = { m u I u € G} ;
 (ii) G[mJ = t u € G I mu = 0} ;
 (iii) G(p) = { u € G I l ui = pn for some n > 0 } ;
 (iv) Gt = { u € G l f ul is finite} .
In particular there are isomorphisms
 (v) Z n[ P] ""' Zp (n > 1 ) and pmzpn f"V Zpn-m (m < n).
 ,

Let H and Gi (i E I) be abelian groups.
 (vi) lfg : G � L Gi is an isomorphism, then the restrictions ofg to mG andG[m]
respectively are isomorphisms mG f"V L mGi and G[m) :=::: .E Gi[m).
 icl


 U.I icl
 (vii) Iff : G � H is an isomorphism, then the restrictions of f to Gt and O(p) re-
spectively are isomorphisms Gt l"'oV Ht and G(p) f"V H(p).

 T

13. Theorem 2.6（定理）

Theorem 2.6. Let G be a finitely generated abelian group.
 (i) There is a unique nonnegative integer s such that the number of infinite cyclic
summands in any decomposition ofG as a direct sum ofcyclic groups is precisely s;
 (ii) either G is free abelian or there is a unique list of (not necessarily distinct)
positive integers mh . . . , mt such that m 1 > 1 , m. I m2 l · · · I mt and


with F free abelian;
 (iii) either G is free abelian or there is a list of positive integerj Pt81 , • , Pk8k,
 • •




which is unique except for the order of its· members, such that p 1 , . . . , Pk are (not
necessarily distinct) primes, s1, . . . , Sk are (not necessarily distinct) positive integers
and


with F free abelian .

14. Corollary 2.7（推论）

Corollary 2.7. Two finitely generated abelian groups G and H are isomorphic if and
only i/G!Gt and H!Ht have the same rank and G and H have the same invariant
factors [resp. elementary divisors].

II.3 The Krull-Schmidt Theorem

15. Definition 3.1（定义）

Definition 3.1. A group G 1sindecomposable ifG :¢ (e) and G is not the (internal)
direct product of two of its proper subgroups.

 Thus G is indecomposable if and only if G :¢ (e) and G "'-' H X K implies
H = (e) or K = (e) (Exercise 1 ).

 EXAMPLES. Every simple group (for example, An, n :¢ 4) is indecomposable.
However indecomposable groups need not be simple : Z, Zpn (p prime) and Sn are in­
decomposable but not simple (Exercises 2 and 1.8.1 ).

16. Definition 3.2（定义）

Defin ition 3.2. A group G is said to satisfy the ascending chain condition (ACC) on
[norntal] subgroups iffor every chain G1 < G2 < · · · of [normal] subgroups of G there
is an integer n such that Gi = Grrfor all i > n. G is said to satisfy the descending chain
condition (DCC) on [normal] subgroups iffor every chain Gt > G2 > · · · of [normal]
subgroups ofG there is an integer n such that Gi = Gn for all i > n.

 EXAMPLES. Every finite group satisfies both chain conditions. Z satisfies the
ascending but not the descending chain condition (Exercise 5) and Z(p ) satisfies00



the descending but not the ascending chain condition (Exercise 1 3).

17. Theorem 3.3（定理）

Theorem 3.3. Ifa group G satisfies either the ascending or descending chain condition
on normal subgroups, then G is the direct product ofafinite number ofindecomposable
subgroups .

18. Lemma 3.4（引理）

Lem ma 3.4. Let G be a group that satisfies the ascending [resp. descending] chain
condition on normal subgroups and f a [norma/J endomorphism ofG. Then f is an auto­
morphism ifand only iff is an epimorphism [resp. n1onomorphism].

19. Lemma 3.5（引理）

Lem ma 3.5. (Fitting) IJG is a group that satisfies both the ascending and descending
chain conditions on normal subgroups and f is a normal endomorphisnJ ofG, then for
some n > I , G = Kel' fn X lm fn.
 I

 J

20. Corollary 3.6（推论）

Corollary 3.6. IJG is an indecomposable group that satisfies both the ascending and
descending chain conditions on normal subgroups and f is a normal endomorphism ofG,
then either f is nilpotent or f is an automorphism.

21. Corollary 3.7（推论）

Corollary 3.7. Let G ( � (e)) be an i11decon1posable group that satisfie�· both the as­
cending and descending chain conditions on normal subgroups. I[ft, . . . , fn are normal
nilpotent endomorph isms ofG such that every fi 1 + · · · + fir ( 1 < it < i 2 < · · · < ir < n)
is an endomorphism, then f1 + f2 + · · · + fn is nilpotent.

22. Theorem 3.8（定理）

Theorem 3.8. (Kru/1-Schmidt) Let G be a group that satisfies both the ascending and
descending chain conditions on normal subgroups. lf G = G1 X G2 X · · · X Ge and
G = H1 X H2 X · · · X Ht with each GhHj indecomposable, then s = t and after
reindexing Gi ,...._, Hi for every i and for each r < t.
 G = G1 X · · · X Gr X Hr+1 X · · · X Ht .

 REMARKS. G has at least one such decomposition by Theorem 3.3. The unique­
ness statement here is stronger than simply saying that the indecomposable factors
are determined up to isomorphism.

II.4 The Action of a Group on a Set

23. Definition 4.1（定义）

Definition 4.1 . An action of a group G on a set S is a function G x S � S
(usually denoted by (g,x) � gx) such that for all x E S and g.,g2 E G:
 ex = x and (g1g2)x = g1(g2x).
When such an action is given, we say that G acts on the set S .
 Since there may be many different actions of a group G on a given set S, the nota­
tion gx is ambiguous. In t:ontext, however, this will not cause any difficulty.

 EXAMPLE. An action of the symmetric group S n on the set In = { 1 ,2 . . . . , n }
is given by ( u,x) � o{x) .

 EXAMPLES. Let G be a group and H a subgroup. An action of the group H on
the set G is given by (h,x) � hx, where hx is the product in G. The action of h E H on
G is called a (left) translation. If K is another subgroup of G and S is the set of all left
cosets of K in G, then H acts on S by translation: (h,xK) � hxK.

 EXAMPLES. Let H be a subgroup of a group G. An action of H on the set G is
given by (h,x) � hxh-1 ; to avoid confusion with the product in G, this action of h E H
is always denoted hxh-1 and not hx. This action of h E H on G is called conjugation by
 1.
 J




h and the element hxh-1 is said to be a conjugate of x. If K is any subgroup of G and
h E H, then hKJz- 1 is a subgroup of G isomorphic to K (Exercise 1.5.6). Hence H acts
on the set S of all subgroups of G by conjugation: (h,K) � hKh-1 • The group hKh�1 is
said to be conjugate to K.

24. Theorem 4.2（定理）

Theorem 4.2. Let G be a group that acts on a set S.
 (i) The relation on S defined by
 x I"J x' {:::::} gx = x' for some g e G
is an equivalence relation.
 (ii) For each x E S, Gx = { g E G I gx = x } is a subgroup of G.

25. Theorem 4.3（定理）

Theorem 4.3. If a group G acts on a set S, then the cardinal number of the orbit of
X E S is the index [G : G ] x .

26. Corollary 4.4（推论）

Corolla ry 4.4. Let G be a finite group and K a subgroup ofG.

 (i) The number of elements in the conjugacy class of x E G is [G : CG(x)], which
divides IGI ;
 (ii) i/Xt, • . . , Xn (xi E G) are the distinct conjugacy classes ofG, then

 3'This agrees with our previous use of the term orbit in the proof of Theorem 1.6.3, where
the special case of a cyclic subgroup (u) of S"' acting on the set In was considered.


 n

 /GI = L [G : Co (xi)] ;
 i=l

 (iii) the number ofsubgroups ofG conjugate to K is [G : No (K)], which divides IGI.

27. Theorem 4.5（定理）

Theorem 4.5. If a group G acts on a set S, then this actio'! induces a homomorphism
G � A(S), where A(S) is the group of all permutations of S.

28. Corollary 4.6（推论）

Corollary 4.6. (Cayley) lfG is a group, then there is a monomorphism G � A(G).
Hence every group is isomorphic to a group ofpermutations. In particular every finite
group is isomorphic to a subgroup ofSn with n = IG/ .

29. Corollary 4.7（推论）

Corollary 4.7. Let G be a group.
 (i) For each g E G, conjugation by g induces an automorphism ofG.
 (ii} There is a homomorphism G --4 A ut G whose kernel is C(G) = { g E G I gx =
xg for all x E G l .

30. Proposition 4.8（命题/性质）

Proposition 4.8. Let H be a subgroup of a group G and let G act on the set S ofall
left cosets ofH in G by left translation. Then the kernel of the induced homomorphism
G ---) A(S) is contained in H.

31. Corollary 4.9（推论）

Corollary 4.9. /fH is a subgroup of index n in a group G and no nontrivial normal
subgroup ofG is contained itt H, then G is isomorphic to a subgroup ofSn.

32. Corollary 4.10（推论）

Corollary 4.10. lf H is a subgroup ofa finite group G ofindex p, where p is the small­
est prime dividing the order ofG, then H is normal in G.

II.5 The Sylow Theorems

33. Lemma 5.1（引理）

Lemma 5.1. If a group H of order pn (p prime) acts on a finite set S and if
So = { x e S I hx = x for all h e H), then l S I = ISol (mod p).

 REMARK. This lemma (and the notation S0) will be used frequently in the

34. Theorem 5.2（定理）

Theorem 5.2. (Cauchy) lfG is a finite group whose order is divisible by a prime p,
then G contains an element of order p.

35. Corollary 5.3（推论）

Corol lary 5.3. A finite group G is a p-group if and only if iGI is a power ofp.

36. Corollary 5.4（推论）

Corol la ry 5.4. The center C(G) of a nontrivial finite p-group G contains more than
one element.

37. Lemma 5.5（引理）

Lem ma 5.5. If H is a p-subgroup of a finite group G, then [No (H) : H] == [G : H]
(mod p).

38. Corollary 5.6（推论）

Corollary 5.6. /f H is p-subgroup ofafinite group G such that p divides [G : H], then
Na(H) � H.

39. Theorem 5.7（定理）

Theorem 5.7. (First Sylow Theorem) Let G be a group of order p nm, with n > 1 , p
prime, and (p,m) = 1 . Then G contains a subgroup oforder pi for each 1 < i < n and
every subgroup of G of order pi (i < n) is normal in some �ubgroup of order pH 1 •

40. Corollary 5.8（推论）

Corol lary 5.8. Let G be a group oforder pnm with p prime, n > 1 and (m,p) = l . Let
H be a p-subgroup ofG.
 (i) H is a Sylow p -subgroup ofG if and only if I H I = pn .
 (ii) Every conjugate of a Sylow p-subgroup is a Sylow p-subgroup.
 (iii) If there is only one Sylow p-subgroup P, then P is normal in G.

41. Theorem 5.9（定理）

Theorem 5.9. (Second Sylow Theorem) lfH is a p-subgroup ofa finite group G, and
P is any Sylow p-subgroup ofG, then there exists x e G such that H < xPx-1 • In par­
ticular, any two Sylow p-subgroups ofG are conjugate.

42. Theorem 5.10（定理）

Theorem 5. 10. (Third Sylow Theorem) JfG is a finite group and p a prime, then the
nwnber of Sylow p-subgroups of G divides IGI and is of the form kp + 1 for some
k > 0.

43. Theorem 5.11（定理）

Theorem 5. 11. If P is a Sylow p-subgroup of a finite group G, then NG(NG(P))
= No(P).

II.6 Classification of Finite Groups

44. Proposition 6.1（命题/性质）

Pro position 6.1. Let p and q be prin1es such that p > q. If q1'p - 1 , then every
group oforder pq is isomorphic to the cyclic group z!JQ • Ifq I p - 1 ' then there are (up
 j



to iso1norphisn1) exactly two distinct groups oforder pq: the cyclic group Zpq and a non­
abelian group K generated by elements c and d such that
 l ei = p ; ldl = q ;
where s 1:- 1 (n1od p) and s q == 1 (mod p).

45. Corollary 6.2（推论）

Corollary 6.2. /fp is an odd prime, then every group oforder 2p is isomorphic either
to the cyclic group Z2v or the dihedral group Dp.

46. Proposition 6.3（命题/性质）

Pro position 6.3. There are (up to isomorphism) exactly two distinct nonabelian
groups of order 8 : the quaternion group Qs and the dihedral group D4 •

 REMARK. The quaternion group Qs is described i n Exercise 1.2.3.

47. Proposition 6.4（命题/性质）

Pro po sition 6.4. There are (up to isomorphism) exactly three distinct nonabelian
groups of order 1 2 : the dihedral group D6 , the alternating group A4, and a group T
generated by elen1ents a,b such that / a I = 6, b2 = as, and ba = a-1b .

II.7 Nilpotent and Solvable Groups

48. Definition 7.1（定义）

Defin ition 7 .1. A group G is nilpotent ifC (G) n = G for some n.

 Every abelian group G is nilpotent since G = C( G) = C1( G).


Theorem 7 2 Every finite p-group is nilpotent.
 . .

49. Theorem 7.3（定理）

Theorem 7 .3. The direct product of a finite nun1ber of nilpotent groups is nilpotent.

50. Lemma 7.4（引理）

Lem ma 7.4. If H is a proper subgroup ofa nilpotent group G, then H is a proper sub­
group of its norntalizer No(H).

51. Proposition 7.5（命题/性质）

Proposition 7.5. A finite group is nilpotent ifand only ifit is the direct product ofits
Sylow subgroups.

52. Corollary 7.6（推论）

Corol lary 7.6. lfG is a finite nilpotent group and m divides /GJ, then G has a sub­
group of order m.

53. Definition 7.7（定义）

Defi n ition 7.7. Let G be a group. The subgroup of G generated by the set
{ aba - tb-1 J a,b c G J is called the commutator subgroup ofG and denoted G'.

 The elements aba-1b-1 (a,b c G) are called commutators. The commutators only
generate G' , so that G' may well contain elements that are not commutators. G is
abelian if and o nly if G' = (e). I n a sense, G' provides a measure of how much G
differs from an abelian group.

54. Theorem 7.8（定理）

Theorem 7 .8. lfG is a group, then G' is a normal subgroup ofG and G/G' is abelian.
lfN is a normal subgroup ofG, then G/N is abelian if and only ifN contains G' .

55. Definition 7.9（定义）

Defin ition 7.9. A group G is said to be solvable if G < n ) = (e) for some n.

 Every abelian group is trivially solvable. More generally, we have

56. Proposition 7.10（命题/性质）

Proposition 7 .10. Every nilpotent group is solvable .

57. Theorem 7.11（定理）

Theorem 7.11. ( i) Every subgroup and every hon1omorphic image ofa solvable group
is solvable.
 (ii) IfN is a norn1al subgroup ofa group G such that N and GjN are solvable, then
G is solvable.

58. Corollary 7.12（推论）

Corol lary 7 . 12. If n > 5, then the symmetric group Sn is not solvable.

59. Lemma 7.13（引理）

Lem ma 7.13. Let N be a normal subgroup of a finite group G and H any sub­
group ofG.
 (i) IJH is a characteristic subgroup ofN, then H is norn1al in G.



 (ii) Every nonnal Sylow p-subgroup ofG is ful(v int·ariant.
 (iii) lfG is so/cable and N is a n1ini11tal nor�nal subgroup. then N is an abelian p­
group for so111e pri111e p .

60. Proposition 7.14（命题/性质）

Proposition 7 . 14. (P. Hall) Ler G be a finite soh·able group of order mn, with
(m,n) = I . Then

 (i) G contains a subgroup oforder m ;
 (ii) any rwo subgroups ofG of order m are conjugate:
 (iii) any subgroup of G of order k, where k I m, is contained in a subgroup of
order m .

 REMARKS. If 111 is a prime power, this theorem merely restates several results
contained in the Sylow theorenls. P. H a l l has also proved the converse of (i) : if G is a
finite group such that whenever I Gl 1nn with (n1,n) = I , G has a subgroup of order
 =



111, then G is solvable. The proof is beyond the scope of this book (see M . Hal l [ 1 5 ;
p. 143]).

II.8 Normal and Subnormal Series

61. Definition 8.1（定义）

Defin ition 8.1. A subnormal series ofa group G is a chain ofsubgroups G = Go >
G1 > · · · > Gn such that Gi+l is normal in Gi for 0 < i < n. The factors of the series
are the quotient groups GJGi+l· The length ofthe series is the number ofstrict inclu­
sions (or alternatively, the number ofnonidentity factors). A subnormal series such that
Gi is normal in G for all i is said to be normal.�

 A subnormal series need not be normal (Exercise 1.5.1 0).

 EXAMPLES. The derived series G > G< 1> > · · > G< n > is a normal series for
 -




any group G (see Exercise 7 . 1 3). If G is nilpotent, the ascending central series
C1(G) < · · · < Cn(G) = G is a normal series for G.

62. Definition 8.2（定义）

Defin ition 8.2. Let G = Go > Gt > · · · > Gn be a subnormal series . A one-step re­
finement ofthis series is any series ofthe form G = Go > · · · > Gi > N > Gi+l > · · ·

 6Some authors use the terms unormal" where we use usubnormal. ••



> Gu or G = Go > · · · > Gn > N, where N is a normal s�bgroup ofGi and (ifi < n)
Gi+l is normal in N. A refinement of a subnormal series S is any subnormal series ob­
tainedfrom S by a finite sequence of one-step refinements. A refinement ofS is said to
be proper if its length is larger than the length ofS.

63. Definition 8.3（定义）

Defin ition 8.3. A subnormal series G = Go > G 1 > · · · > Gn = (e ) is a composi­
tion series ifeach factor Gi/Gi+l is simple. A subnormal series G = Go > G 1 > · · · >
Gn = (e) is a solvable series if each factor is abelian.

 The following fact is used frequently when dealing with composition series : if N is
a normal subgroup of a group G, then every normal subgroup of GIN is of the form
HI N where H is a normal subgroup of G which contains N (Corollary 1 . 5 . 1 2) . There­
fore, when G � N, GIN is simple if and only if N is a maximal in the set of all
normal subgroups M of G with M � G (such a subgroup N is called a maximal
normal subgroup of G).

64. Theorem 8.4（定理）

Theorem 8.4. (i) Every finite group G has a composition series.
 (ii) Every refinement of a solvable series is a solvable series.
 (iii) A subnormal series is a co1nposition series if and only if it has no proper re­
finements.

65. Theorem 8.5（定理）

Theore m 8.5. A gruup G is solvable if and only if it has a solvable series.

66. Proposition 8.6（命题/性质）

Proposition 8.6. A finite group G is solvable ifand only ifG has a composition series
whose factors are cyclic ofprime order.

67. Definition 8.7（定义）

Defin ition 8.7. Two subnormal series S and T ofa group G are equivalent ifthere is a
one-to-one correspondence between the nontrivial factors ofS and the nontrivial factors
ofT such that corresponding factors are isomorphic groups.

 Two subnormal series need not have the same number of terms in order to be
equivalent, but they must have the same length (that is, the same number of non­
trivial factors). Clearly, equivalence of subnormal series is an equivalence relation.

68. Lemma 8.8（引理）

Lem ma 8.8. lfS is a composition series of a group G, then any refinement ofS is
equivalent to S.

69. Lemma 8.9（引理）

Lem ma 8.9. (Zassenhaus) Let A*, A, B *, B be subgroups of a group G such that A*
is normal in A and B* is normal in B.



 (i) A *(A n B *) is a normal subgroup ofA *(A n B) ;
 (ii) B*(A * n B) is a normal subgroup of B*(A n B);
 (iii) A *(A n B)/ A *(A n B*) r-v B *(A n B)/B*(A * n B).

70. Theorem 8.10（定理）

Theorem 8.10. (Schreier) Any two subnormal [resp. normal] series ofa group G have
subnormal [resp. normal] refinements that are equivalent.

71. Theorem 8.11（定理）

Theorem 8.11. (Jordan-Holder) Any two composition series of a group G are
equivalent. Therefore every group having a composition series determines a unique list
of sbnple groups.

 REMARK. The theorem does not state the existence of a composition series for a
given group.

Chapter III: Rings

III.1 Rings and Homomorphisms

72. Definition 1.1（定义）

Defin ition 1.1. A ring is a nonempty set R together with two binary operations
(usually denoted as addition ( +) and multiplication) such that:
 (i) (R,+) is an abelian group;
 (ii) (ab)c = a(bc) for all a,b,c e R (associative multiplication);
 (iii) a(b + c) = ab + ac and (a + b)c = ac + be (left and right distributive
laws).
/fin addition:
 (iv) ab = ba for all a,b e R,
then R is said to be a commutative ring. IfR contains an element lR such that
 (v) IRa = alR = a for all a e R,
then R is said to be a ring with identity.

 REMARK. The symbol 1 R is also used to denote the identity map R --+ R. In
context this usage will not be ambiguous.

 The additive identity element of a ring is called the zero element and denoted 0.
If R is a ring, a e R and n e Z, then na has its usual meaning for additive groups
(Definition 1.1 .8) ; for example, na = a + a + · · · + a (n summands) when n > 0.
Before giving examples of rings we record

73. Theorem 1.2（定理）

Theorem 1.2. Let R be a ring. Then
 (i) Oa = aO = O for all a e R ;
 (ii) ( - a)b = a( - b) = - (ab) for all a,b e R ;
 (iii) ( - a)(- b) = ab for all a,b e R ;
 (iv) (na)b = a(nb) = n(ab) for all n e Z and all a,b e R ;

 (v) (t ) (f ) f f
 t=l
 a;
 J=l
 b; =
 t= l J = l
 a ; b; for all a;,b; E R .

74. Definition 1.3（定义）

Defin ition 1.3. A nonzero element a in a ring R is said to be a left [resp. right] zero
divisor if there exists a nonzero b e R such that ab = 0 [resp. ba = 0]. A zero divisor
is an element ofR which is both a left and a right zero divisor.

 It is easy to verify that a ring R has no zero divisors if and only if the right and
left cancellation laws hold in R ; that is, for all a,b,c e R with a � 0,
 ab = ac or ba = ca b = c.

75. Definition 1.4（定义）

Defin ition 1.4. An element a in a ring R with identity is said to be left [resp. right] in­
vertible if there exists c e R [resp. b e R] such that ca = lR [resp. ab = ln]. The ele­
ment c [resp. b) is called a left [resp. right] inverse ofa. An element a e R that is both
left and right invertible is said to be invertible or to be a unit.

 REMARKS. (i) The left and right inverses of a unit a in a ring R with identity
necessarily coincide (since ab = 1 R = ca implies b = 1 Rb = (ca)b = c(ab) = c1 R = c).
(ii) The set of units in a ring R with identity forms a group under multiplicat ion .

76. Definition 1.5（定义）

Defin ition 1.5. A commutative ring R with identity lu � 0 and no zero divisors is
called an integral domain. A ring D with identity l D � 0 in which every nonzero ele­
ment is a unit is called a division ring. A field is a commutative division ring.

 REMARKS. (i) Every integral domain and every division ring has at least two
elements (namely 0 and 1 R). (i i) A ring R w ith identity is a division ring if and only if
the nonzero elements of R form a grou p under multi plication (see R emark (ii) a fter
imply that b = 1 Fb = (a-1a)b = a-1(ab) = a-1 0 = 0.

 EXAMPLES. Th e ring Z of i ntegers is an integral domain. The set E of even
integers is a commutative ring without identity. Each of Q (rationals), R (real
numbers), and C (complex numbers) is a field under the usua l operat ions of addition
and multipli cat ion. The n X n matrices over Q (or R or C) form a noncommutative
ring with identity. The units i n this ring are precisely the nonsingular n1atrices.

 EXA:\1PLE. For each positive integer n the set Z,, of integers modulo n is a ring.
See the exarnple after Theorem 1 . 1 .5 for details. I f n is not prime, sa y n = kr with
k > 1 , r > 1 , then i: � 0, r � 0 and k r = kr = ii = 0 i n Zn , whence k and r are
zero divisors. If p is prime, then Zr is a field by E xercise 1 . 1 .7.

 EXA1\IPLE. Let A be an abelian group and let End A be the set of endomor­
phisms f : A - � A . Define addition in End A by ( f + g)(a) = /(a) + g(a). Verify
that f + g � End A . Since A is abelian, this makes End A an abel ian group. Let multi­
plication in End A be given by composition of functions. Then End A is a (possibly
noncommutative) ring with identity l _t : A A.__...




 L
 EXA,IPLE. Let G be a (multi plicative) group and R a ring. Let R( G) be the
additive abelian group R (one copy of R for each g ; G). It will be convenient to
 yEG



adopt a new notation for the elements of R( G). An element x = { r0 } or.G of R( G) has
only finitely many nonzero coordinates, say r0u . . . , r0n (gi e G). Denote x by the
 n

formal sum r01gt + Tg2g2 + · · · + r011gn or L r0igi. We also allow the possibility that
 i=l
some of the rg, are zero or that some gi are repeated, so that an element of R(G)
may be written in formally different ways (for example, r1g1 + Og2 = r1g1 or
r1g1 + s1g1 = {r1 + s1) g1). In this notation, addition in the group R (G) is given by:
 n n n


 L ro igi + L Soigi = L (roi + Soi)gi ;
 i-1 i= l i=l

(by inserting zero coefficients if necessary we can always assume that two formal
sums involve exactly the same indices g., . . . , gn). Define multiplication in R( G) by



this makes sense since there is a product defined in both R (r;si) and G (g; h) and thus
the expression on the right is a formal sum as desired. With these operations R( G) is
a ring, called the group ring of G over R. R( G) is commutative if and only if both R
and G are commutative. If R has an identity 1 R, and e is the identity element of G,
then 1 Re is the identity element of R( G).

 EXAMPLE. Let R be the field of real numbers and S the set of symbols l ,i,j,k.
Let K be the additive abelian group R EB R E8 R EB R and write the elements of K as
formal sums (ao,ax,a2,aa) aol + a.i + a0 + aak. Then aol + a.i + a2j + aak =
 =



b01 + bd + h2j + bak if and only if ai hi for every i. We adopt the conventions
 =



that aol e K is identified with ao e R and that terms with zero coefficients may be
omitted (for example, 4 + 2j = 4 · 1 + Oi + 2j + Ok and i = 0 + l i + Oj + Ok) .
Then addition in K is given by
 (ao + ad + aJ + aak) + (bo + bd + b2j + bak)
 = (ao + bo) + (a1 + b1)i + (a2 + b2)j + (aa + b3)k .

Define multiplication in K by
 (ao + ad + aJ + aak)(bo + bti + b2j + bak)
 = (aobo - axbt - a2h2 - aab3) + (aobi + a1bo + a2ba - aab2) i
 + (aob'l. + a2bo + aabt - a1ba)j + (aoba + aabo + a1b2 - a2b1)k.
This product formula is obtained by multiplying the formal sums term by term sub­
ject to the following relations : (i) associativity; (ii) ri = ir; rj = jr, rk = kr (for all
r s R) ; (iii) P= j2 = k2 = ijk = - 1 ; ij = -ji = k; jk = - kj = i; ki = - ik = j.
Under this product K is a noncommutative division ring in which the multiplicative
inverse of at) + ad + azj + a3k is (a0/d) - (a1/d)i - (a2jd)j - (a3/d)k, where
d ao2 + a12 + a-l + a�?·. K is called the division ring of real quaternions. The
 =



matrices over the field C of complex numbers (E xercise 8).
 Defi n ition 1 . 1 shows that under multiplication the elements of a ring R form a
semi group (a monoid if R has an identity). Consequently Definition I. t .8 is appli­
cable and exponentiation is defined in R . We have for each a e R and n e N*,



Subtraction in a r i n g R is defined in the usual way: a - = + ( - b). Clearly b a
a(b- c) = ab
 - ac and (a - b)c
 = ac - for all € R. be a,b,c
 The next theorem is frequently useful in computations. Recall that if k and n are
integers with 0 < k < n, then the binomial coefficient (Z) is the number
n !/(n - k)!k !, where 0 ! = 1 and n ! n(n l )(n 2) · · 2 · 1 for n >
 = - - (%) is · 1.

a ctually an integer (Exercise 1 0).

77. Theorem 1.6（定理）

Theorem 1.6. (Binomial Theorem). Let R be a ring with identity, n a positive integer,
and a,b,a1,a2, . . . , ae R.e


 (i) /fab ba, then (a + b)n .t (�) akbn-k ;
 = =
 k=O -.




where the sum is over all s-tup/es (it,i2, . . . , i�) such that it + i 2 + · · · + if\ = n .

78. Definition 1.7（定义）

Definition 1.7. Let R and S be rings. A function f : R � S is a homomorphism of
rings provided that for all a,b R :
 e

 f(a + b) = f(a) + f(b) and f(ab) = f(a)f(b).

 REMARK. It is easy to see that the class of all rings together with all ring homo­
morphisms forms a (concrete) category.
 When the context is clear then we sha l l frequently write "homomorphism. , i n
pl ace of "homomorphism of rings." A homomorphism of rings is, in part icular, a
homomorphism of the underlying additive grou ps. Consequently the same termi­
nology is used : a monomorphism [resp . epimorphism , isomorphism] of rings is a homo-



morphism of rings which is an injective [resp. surjective, bijective] map. A mono­
morphism of rings R --) S is sometimes cal1ed an embedding of R in S. An isomor­
phism R --.. R is called an automorphism of R.
 The kernel of a homomorphism of rings f : R --) S is its kernel as a map of addi­
tive groups ; that is, Ke r f = { r e R I f(r) = 0 } . Similarly the i mage of f, denoted
lm f, is { s e S I s = f(r) for some r e "R } . If R and S both have identities lR and 1 8, we
do not require that a homomorphism of rings map l R to 18 (see Exercises 15, 16).

 Zm given by k � k is an epimorphism of
 EXAMPLES. The canonical map- Z --..-
rings. The map Z3 � Z6 given by k � 4k is a well-defined monomorphism of
.

nngs.

 EXAMPLE. Let G and H be multiplicative groups and f : G � H a homomor­
phism of groups. Let R be a ring and define a map on the group rings 1 : R( G)-) R(H)
by:




Then 1 is a homomorphism of rings.



 .. ..



for all a z R, then R is said to have characteristic n. If no such n exists R is said to
have characteristic zero. (Notation : char R = n).

79. Theorem 1.9（定理）

Theorem 1.9. Let R be a ring with identity J R and characreristic n > 0.
 (i) If cp : Z � R is the map given by m � mlR, then cp is a homomorphism of
rings with kernel (n) = I kn I k e Z} .
 ( ii) n is the least posith·e integer such that n IR = 0 .
 (iii) If R has no zero divisors (in particular ifR is an integral domain), then n is
prune.

80. Theorem 1.10（定理）

Theorem 1.10. Every ring R nutJ' be en1bedded in a ring S with identity. The ring S
(which is not unique) 1nay be chosen to be either of characteristic zero or of the same
characteristic as R .

III.2 Ideals

81. Definition 2.1（定义）

Def i n ition 2.1. Let R he a ring and S a nonen1pty subset ofR that is closed under the
operations ofaddition and Jnultiplication in R. lf S is itselfa ring under these operations
then S is called a subring of R . A subring I of a ring R is a left ideal provided
 reR and x E l rx c I ;
I is a right ideal provided
 r e R and x e l xr e I ;
I is an ideal if it is both a left and right ideal.

 Whenever a statement is made about left ideals it is to be understood that the
analogous statement holds for right ideals.

 EXAMPLE. If R is any ring, then the center of R is the set C =
 { c E R I cr = rc
for all r z R } . C is easily seen to be a subring of R. but may not be an ideal (Exer­
cise 6).

 EXAI\1PLE. If f : R ---Jo S is a homomorphisn1 of rings, then Ker fis an ideal in R
(Theorem 2 . 8 below) and Im f is a subring of 5. lm f need not be an ideal in S.

 EXAI\1PLE. For each integer n the cyclic subgroup (n ) = { kn I k E Z } is an

 J
idea l in Z.

 EXAfVIPLE. I n the ring R of 11 X n n1 atrices over a division ring D, let h· be the
set of all matrices that ha ve nonzero entries only in column k . Then h is a left ideal,

 j



but not a right ideaL lf Jk consist of those matrices with nonzero entries only in row
k then Jk is a right ideal but not a left ideal.
 �




 EXA\ IPLE. Two ideals of a ring R are R itself and the trivial ideal (denoted 0),
which consists only of the zero element.

 REI\ lARKS. A [left] ideal / of R such that I ¢ 0 and I ¢ R is called a proper [left]
ideal. Observe that if R has an identity l 1l and I is a [left] ideal of R, then I = R if and
only if 1 u ; I. Consequently, a nonzero [left] ideal / of R is proper if and only if I con­
tains no units of R; (for if u z R is a unit and u � 1., then l R = u-1 u E 1). I n particular, a
division ring D has no proper left (or right) ideals since every nonzero element of D is
a unit. For the converse, see Exercise 7. The ring of n X n matrices over a division

ring has proper left and right ideals (see above), but no proper (two-sided) ideals
(Exercise 9).

82. Theorem 2.2（定理）

Theorem 2.2. A nonen1pty subset I ofa ring R is a left [resp. right] ideal ifand only if
for all a,b e I and r � R :

 (i) a,b 2 I => a - b e I ; and
 ( ii ) a z I, r E R � ra ; I [resp. ar e I].

83. Corollary 2.3（推论）

Corol lary 2.3. Let I Ai ! i e I I he a fcn11i/y of [left] ideals in a ring R. Then n Aa is
 iel

also a [l�ft] ideal.

84. Definition 2.4（定义）

Defin ition 2.4. Let X be a subser of a ring R. Let { Ai I i E I J be the Jan1ily of all
[leftJ ideals in R which contain X . Then n Ai is called the [lefr] ideal generated by X.
 icl

This ideal is denoted (X).

 The elements of X are cal led generators of the ideal (X). If X = { xh .Xn } ,
 . . •



then the ideal (X) is denoted by (x � ,x:!, . . . ) and said to be finitely generated. An
 , x,


ideal (x) generated by a single element is called a principal ideal. A principal ideal ring
is a ring in which every ideal is principal. A principal ideal ring which is an integral

85. Theorem 2.5（定理）

Theorem 2.5. Let R be a ring a ; R and X c R .
 (i) The principal ideal (a) consists of all elements of the form ra + as + na +
I riasi (r,s,ri ,si e R;
 m

 m e N*; and n e Z).
1- l




 2The term .. principal ideal ringn is som etimes used in the literature to denote what we
have ca lied a principal ideal domain.




 (ii) IfR has an identity, then (a) = {t
 �= 1
 }
 riasi I r;,si E R; n E N • .
 (iii) /fa is in the center ofR, then (a) = { ra + na I r e R, n e Z} .
 (iv) Ra = { ra I r e R } [resp. aR = { ar I r e R } ] is a left [resp. right] ideal in R
(which may not contain a). lf R has an identity, then a e Ra and a e aR.
 (v) lf R has an identity and a is in the center ofR, then Ra = (a) = aR.
 (vi) lfR has an identity and X is in the center of R, then the ideal (X) consists of
all finite sums r 1a1 + · · + rnan (n e N * ; r i e R ; ai e X).
 ·




 REMARK. The hypothesis of (iii) is always satisfied in a commutative ring.

86. Theorem 2.6（定理）

Theorem 2.6. Let A,A1,A2, . . . , An, B and C be [left] ideals in a ring R .
 (i) At + A2 + · · · + A n and A 1 A 2 · · · A n are [left] ideals;
 (ii) (A + B) + C = A + (B + C) ;
 (iii) (AB)C = ABC = A(BC) ;
 (iv) B(At + A2 + · + An) = BAt + BA2 + · · · BAn ; and (At + A2 + · · · +
 · ·




An)C = AtC + A2C + · · · + AnC.

87. Theorem 2.7（定理）

Theorem 2.7. Let R be a ring and I an ideal ofR . Then the additive quotient group
R/I is a ring with multiplication given by
 (a + l)(b + I) = ab + I.
lfR is commutative or has an identity, then the same is true ofR/I.

88. Theorem 2.8（定理）

Theorem 2.8. Iff : R � S is a homomorphism ofrings, then the kernel off is an ideal
in R. Conversely ifl is an ideal in R, then the map 1r : R � R /1 gil,en by r � r + I is
an epimorphism of rings with kernel I.

 The map 1r is called the canonical epimorphism (or projection).

89. Theorem 2.9（定理）

Theorem 2.9. If f : R � S is a homomorphism of rings and I is an ideal of R rvhich is
contained in the kernel off, then there is a unique homomorphism ofrings f : R/1 � S
such that f(a + I) = f(a) for all a z R . lm 1 = bn f and Ker f = (Ker f)/1. f is an iso­
morphism if and only if f is an epintorphism and I = Ker f.

90. Corollary 2.10（推论）

Corollary 2.10. (First Isomorphism Theorem) If f : R � S is a homomorphism of
rings, then f induces an isomorphism ofrings R/ Ker f "'"' lm f.

91. Theorem 2.12（定理）

Theorem 2.12. Let I and J be ideals in a ring R.
 (i) (Second Isomorphism Theorenz) There is an isomorphisms ofrings 1/(1 n J) "'"'
(I + J)jJ;
 (ii) (Third Isomorphism Theorem) if I C J, then Jjl is an ideal in R/1 and there is
an isomorphism ofrings (R/1)/(1/1) ""' R/1.

92. Theorem 2.13（定理）

Theorem 2.13. If I is an ideal in a ring R , then there is a one-to-one correspondence
between the �et ofall ideals of R which contain I and the set ofall ideals of R/1, giDen
by J J----t 1/1. Hence every ideal in R/I is ofthe form 1/1, where J is an ideal of R which
contains I.

93. Definition 2.14（定义）

Defi nition 2.14. An ideal P in a ring R is said to be prime ifP � R andfor any ideals
A,B in R
 AB c P � A c P or B c P.

 The definition of prime ideal excludes the ideal R for both historical and technical
reasons. Here is a very useful characterization of prime ideals ; other characteriza­
tions are given in Exercise 14.

94. Theorem 2.15（定理）

Theorem 2.15. If P is an ideal in a ring R such that P #- R and for all a,b e R

 ab e P =:) a e P or b e P, (1)
then P is prime. Conversely ifP is prime and R is commutative, then P satisfies con­
dition (1).

 REMARK. Commutativity is necessary for the converse (Exercise 9 (b)).

95. Theorem 2.16（定理）

Theorem 2.16. In a commutative ring R with identity lR ;t. 0 an ideal P is prime
if and only if the quotient ring R/P is an integral domain.

96. Definition 2.17（定义）

Defin ition 2.17. An ideal [resp. left ideal] M in a ring R is said to be maximal if
M � R and for every ideal [resp. left ideal] N such that M C N C R, either N = M
or N = R.

 EXAMPLE. The ideal (3) is maximal in Z; but the ideal (4) is not since (4) c
 #

(2) c z.
 ,e



 REMARK. If R is a ring and S is the set of all ideals I of R such that I � R, then
S is partially ordered by set-theoretic inclusion. M is a maximal ideal (Definition 2.17)
if and only if M is a maximal element in the partially ordered set S in the sense of
Introduction, Section 7. More generally one sometimes speaks of an ideal I that is
maximal with respect to a given property, meaning that under the partial ordering of
set theoretic inclusion, I is maximal in the set of all ideals of R which have the given
property. In this case I need not be maximal in the sense of Definition 2. 1 7 .

97. Theorem 2.18（定理）

Theorem 2.18. In a nonzero ring R with identity maximal [left] ideals always exist.
In fact every [left] ideal in R (except R itself) is contained in a maxinral [left] ideal.

98. Theorem 2.19（定理）

Theorem 2.19. /fR is a commutative ring such that R2 = R (in particular ifR has an •




identity), then every maximal ideal M in R is prbne.

 REMARK. The converse of Theorem 2.1 9 is false. For example, 0 is a prime
 ·



ideal in Z, but not a maximal ideal. See also Exercise 9.

99. Theorem 2.20（定理）

Theorem 2.20. Let M be an ideal in a ring R with identity I R '#- 0.
 I



 l



 (i) JfM is maximal and R is commutative, then the quotient ring R/M is afield.
 (ii) If the quotient ring R/M is a division ring, then M is maximal.

 REMARKS. (i) is false if R does not have an identity (Exercise 19). If M is maxi­
mal and R is not commutative, then R/ M need not be a division ring (Exercise 9).

100. Corollary 2.21（推论）

Corollary 2.21. The following conditions on a commutative ring R with identity
1n ¢. 0 are equivalent.
 (i) R is a field;
 (ii) R has no proper ideals;
 (iii) 0 is a maximal ideal in R ;
 (iv) every nonzero homomorphism of rings R --+ S is a monomorphism.

 REMARK. The analogue of Corollary 2.21 for division rings is false {Exercise 9).

101. Theorem 2.22（定理）

Theorem 2.22. Let { Ri I i E I } be a nonempty family of rings and II Ri the direct
 ial
product of the additive abelian groups Ri;



 (i) II Ri is a ring w ith multiplication defined by ( a d itl { bd h i = { aib d i !l ;
 (ii) ifRi has an identity [resp. is co1nmurative] for every i € I, then II Ri has an
 id
identity [resp. is com1nutative];
 (iii) for each k E I the canonical projection 7rk : II Ri ---+ R k given by { ai l � ak , is
 ie.I
an epimorphism of rings;
 (iv) for each k € I the canonical injection Lk : Rk ---+ II Ri, gicen by a k � ( ai }
(where ai = 0 for i ¢ k), is a monomorphism of rings.

102. Theorem 2.23（定理）

Theorem 2.23. Let { Ri I i e I } be a nonempty Ja1ni/y of rings, S a ring and
( <Pi : S � Ri I i € I } a family ofhomo1norphisms ofrings. Then there is a unique homo­
morphisln of rings <P : S � II R i such that 1ri<P = cpi for all i E I. The ring II Ri is
 id ie.l
uniquely determined up to isomorphism by this property. In other words II R i is a
 icl
product in the category of rings.

103. Theorem 2.24（定理）

Theore m 2.24. Let A 1 ,A2, . . . , A n be ideals in a ring R such that (i) At + A2 + · · · +
A n = R and (ii) for each k (1 < k < n), Ak n (A t + · · + Ak-t + Ak+ l + · + A n)
 · · ·




= 0. Then there is a ring isomorphism R "" At X A2 X · · · X An.

104. Theorem 2.25（定理）

Theorem 2.25. (Chinese Remainder Theorem) Let At, . . , An be ideals in a ring R
 .




such that R2 + Ai R for all i and Ai + Aj = R for all i � j. lf bt, . . . , bn E R,
 =



then there exists b E R such that
 (i = 1 ,2, . . . , n).
Furthermore b is uniquely determined up to congruence modulo the ideal



 REMARK. If R has an identity, then R 2 = R, whence R 2 + A = R for every
ideal A of R.

105. Corollary 2.26（推论）

Corollary 2.26. Let m 1 ,m2 , . . . , mn be positive integers such that (mi,mj) = 1 for
i � j . /f bt ,b2, . . . , bn are any integers, then the system of congruences



has an integral solution that is uniquely determined modulo m = m1m2 · · · fin .
 n

106. Corollary 2.27（推论）

Corollary 2.27. If At, . . . , An are ideals in a ring R, then there is a monomorphism
of rings
 0 : Rj(At n · · n An) � R/ At X R/ A2 X · · · X R/ A
 · •..




/f R 2 + Ai = R for all i and Ai + A1 = R for all i � j, then 0 is an isomorphism
of rings.

III.3 Factorization in Commutative Rings

107. Definition 3.1（定义）

Defin ition 3.1. A nonzero ele1nent a of a commutative ring R is said to divide an
element b E R (notation : a I b) ifthere exists x e R such that ax = b. Elements a,b ofR
are said to be associates if a I b and b I a.

 Virtually all statements about divisibility may be phrased in terms of principal
ideals as we now see.

108. Theorem 3.2（定理）

Theorem 3.2. Let a�b and u be elements of a commutath·e ring R with identity.
 (i) a I b ifand only if (b) C (a).
 (ii) a and b are associates ifand only if(a) = (b).
 (iii) u is a unit ifand only 1ju I r for all r e R.
 (iv) u is a unit ifand only if(u) = R .
 (v) The relation . .a is an associate ofb" is an equivalence relation on R .
 (vi) /f a = br with r e R a unit, then a and b are associates. lf R is an integral
domain, the converse is true.

109. Definition 3.3（定义）

Defin ition 3.3. Let R be a commutative ring with identity. An element c of R ts
irreducible provided that:

 (i) c is a nonzero nonunit;
 (ii) c = ab ==> a or b is a unit.
An element p ofR is prime provided that:
 (i) p is a nonzero nonunit;
 (ii) p I ab :::::} p I a or p I b.

 EXAMPLES. If p is an ordinary prime integer, then both p and -p are irre­
ducible and prime in Z in the sense of Definition 3.3. In the ring Zth 2 is easily seen to
be a prime. However 2 e Z6 is not irreducible since 2 = 2 4 and neither 2 nor 4 are
 ·




units in Z6 (indeed they are zero divisors). For an example of an irreducible element
which is not prime, see Exercise 3.
 There is a close connection between prime [resp. irreducible] elements in a ring R
and prime [resp. maximal] principal ideals in R.

110. Theorem 3.4（定理）

Theorem 3.4. Let p and c be nonzero elements in an integral domain R .
 (i) p is prime ifJnd only if(p) is nonzero prime ideal;
 (ii) c is irreducible ifand only if(c) is maximal in the set-S ofall proper principal
ideals ofR.
 (iii) Every prime element of R is irreducible.
 (iv) lfR is a principal ideal domain, then p is prime ifand only ifp is irreducible.
 (v) Every associate of an irreducible [resp. prime] element of R is irreducible
[resp. prime].
 (vi) The only divisors of an irreducible element of R are ·its associates and rhe
units ofR.

 REMARK. Severa] parts of Theorem 3.4 are true for any commutative ring with
identity, as is seen in the following proof.

111. Definition 3.5（定义）

Definition 3.5. An integral domain R is a unique factorization domain provided that:
 (i) every nonzero non unit element a of R can be written a = C1C2 · Cn, with · ·




c., . . . , Cn irreducible.
 (ii) If a = C1C2 · · · Cn and a = d1 d2 drn (ci,di irreducible), then n = m and for
 · · ·




some permutation a of { 1 ,2, . . . , n } , Ci and du<o are associates for every i.

 REMARK. Every irreducible element in a unique factorization domain is neces­
sarily prime by (ii). Consequently, irreducible and prime elements coincide by
Theorem 3.4 (iii).


 Definition 3.5 is nontrivial in the sense that there are integral domains in which
every element is a finite product of irreducible elements, but this factorization is not
unique (that is, Definition 3.5 (ii) fails to hold) ; see Exercise 4. Indeed one of the
historical reasons for introducing the concept of ideal was to obtain some sort of
unique factorization theorems (for ideals) in rings of algebraic integers in which
 In view of the relationship between irreducible elements and principal ideals
(Theoretn 3.4) and the example of the integers, it seems plausible that every principal
ideal domain is a unique factorization domain. In order to prove that this is indeed
the case we need :

112. Lemma 3.6（引理）

Lem ma 3.6. lfR is a principal ideal ring and (a.) C (a2) C · · · is a chain ofideals in
R, then for some positive integer n, (aj) = (an) for all j > n.

113. Theorem 3.7（定理）

Theorem 3.7. Every principal ideal domain R is a unique factorization domain.

 REMARK. The converse of Theorem 3.7 is false. For example the polynomial
ring Z[x] can be shown to be a unique factorization domain (Theorem 6.14 below),
but Z[x] is not a principal ideal domain (Exercise 6. 1 ). .

114. Definition 3.8（定义）

Defin ition 3.8. Let N be the set ofnonnegative integers and R a commutative ring.
R is a Euclidean ring if there is a function cp : R - { 0} � N such that:
 (i) ifa,b e R and ab =I= 0, then �a) < {O(ab);
 (ii) ifa,b E R and b � 0, then there exist q,r e R such that a = qb + r with r = 0,
 or r � 0 and 'P(r) < tp{b).
A Euclidean ·ing which is an integral domain is called a Euclidean domain.

 EXAMPLE. The ring Z of integers with cp(x) = lxl is a Euclidean domain .

 EXAMPLE. If F is a field, let cp(x) = 1 for all x e F, x � 0. Then F is a Euclidean
domain.

 EXAMPLE. If F is a field, then the ring of polynomials in one variable F[x] is a
Euclidean domain with cp( f) = degree of f; see Corollary 6.4 below.

 EXAMPLE. Let Z[/J be the following subset of the complex numbers
Z[i) = ( a + hi 1 a, b e Z } . Z(iJ is an integral domain called the domain of Gaussian
integers. Define <P(a + bi) = a2 + b2 • Clearly cp(a + hi) � 0 if a + hi � 0; it is also
easy to show that condition (i) of the definition is satisfied. The proof that cp satisfies
condition (ii) is left to the reader (Exercise 6).

115. Theorem 3.9（定理）

Theorem 3.9. Every Euclidean ring R is a principal ideal ring with identity. Con­
sequently every Euclidean domain is a unique factorization domain.

 REMARK. The converse of Theorem 3 .9 is false since there are principal ideal
domains that are not Euclidean domains (Exercise 8).

116. Definition 3.10（定义）

Definition 3.10. Let X be a nonempty subset of a commutative ring R. An element
d E R is a greatest common divisor of X provided:
 (i) d I a for all a E X ;
 (ii) c I a for all a E X ==> c I d.

 Greatest common divisors do not always exist. For example, in the ring E of even
integers 2 has no divisors at all, whence 2 and 4 have no (greatest) common divisor.
Even when a greatest common divisor of a1 , . . . , an exists, it need not be unique.
However, any two greatest common divisors of X are clearly associates by (ii).
Furthermore any associate of a greatest common divisor of X is easily seen to be a
greatest common divisor of X. If R has an identity and a1 ,a2 , , an have l R as a • • •




greatest common divisor, then a. ,a2 , an are said to be relatively prime.
 • • •

117. Theorem 3.11（定理）

Theorem 3.11. Let a., . . . , a be elements of a commutative ring R with identity.
 ..




 (i) d E R is a greatest common divisor of { a 1 , . . . , an J such that d = rial
+ · · · + rnan for some ri E R if and only if(d) (at) + (a2) + · + (an) ; = · ·

 (ii) if R is a principal ideal ring, then a greatest common divisor of a 1 , . . . , a n
exists and every one is of the form r a + i
 + rnan (ri E R) ;
 l · · ·

 (iii) if R is a unique factorization domain, then there exists a greatest common
divisor ofa�, . . . ' an.

 REMARK. Theorem 3. l l (i) does not state that every greatest common divisor of
a�, . . . , an is expressible as a linear combination of a., . . . , a, . In general this is not
the case (Exercise 6.1 5). See also Exercise 1 2.

III.4 Rings of Quotients and Localization

118. Definition 4.1（定义）

Defin ition 4.1. A nonempty subset S of a ring R is multiplicative provided that
 a,b e S � ab e S.

 EXAMPLFS. The set S of all elements in a nonzero ring with identity that are
not zero divisors is multiplicative. In particular, the set of all nonzero elements in an
integral domain is multiplicative. The set of units in any ring with identity is a
multiplicative set. If P is a prime ideal in a commutative ring R, then both P and
S = R - P are multiplicative sets by Theorem 2.15.
 The motivation for what follows may be seen most easily in the ring Z of integers
and the field Q of rational numbers. The set S of all nonzero integers is clearly a
multiplicative subset of Z. Intuitively the field Q is thought of as consisting of all
fractions a/ b with a e Z and b e S, subject to the requirement
 ajb = c/d <=> ad = be (or ad - be = 0).
More precisely, Q may be constructed _as follows (details of the proof will be
supplied later). The relation on the set Z X S defined by
is easily seen to be an equivalence relation. Q is defined to be the set of equivalence
classes of Z X S under this equivalence relation. The equivalence class of (a,b) is
denoted ajb and addition and multiplication are defined in the usual way. One
verifies that these operations are well defined and that Q is a field. The rna p Z � Q
given by a � a/ 1 is easily seen to be a monomorphism (embedding).
 We shall now extend the construction just outlined to an arbitrary multiplicative
subset of any commutative ring R (possibly without identity). We shall construct a
commutative ring s- 1R with identity and a homomorphism cps : R � S-1R. If S is
the set of all nonzero elements in an integral domain R, then s-1R will be a field
(S-1 R = Q if R = Z) and cps will be a monomorphism embedding R in s-1R.

119. Theorem 4.2（定理）

Theorem 4.2. Let S be a multiplicative subset ofa commutative ring R. The relation
defined on the set R X S by
 (r ,s) r-..J (r' ,s') s1 (rs' - r's) = 0 for some s1 e: S !

 I
 �



 i


is an equivalence relation . Furthermore ifR has no zero divisors and 0 ' S, then
 (r.s) --- (r:s') (:=} rs' - r's = 0.

120. Theorem 4.3（定理）

Theorem 4.3. Let S be a multiplicative subset ofa commutative ring R and let s-IR
be the set ofequivalence classes ofR X S under the equivalence relation o/Theorem 4.2.
 (i) S-1R is a commutative ring with identity, where addition and multiplication are
defined by
 r/s + r'/s' = (rs' + r's)/ss' and (rjs)(r'/s') = rr'/ss'.
 (ii) 1/R is a nonzero ring with no zero divisors and 0 ' S, then s-1R is an integral
domain.
 (iii) 1/R is a nonzero ring with no zero divisors and S is the set ofall nonzero ele­
ments ofR , then S-1R is a field.

121. Theorem 4.4（定理）

Theorem 4.4. Let S be a multiplicative subset of a commutative ring R .
 (i) The map <Ps : R � s-IR given by r t---+ rs/s (for any s e S) is a well-defined
homomorphism of rings such that cps(s) is a unit in s-IR for ecery s e S.
 (ii) JfO t S and S contains no zero divisors, then <Ps is a monomorphism. In par­
ticular, any integral do1nain may be en1bedded in its quotient field.
 (iii) /f R has an identity and S consists ofunits, then cps is an isomorphism. In par­
ticular, the co1nplete ring ofquotients ( = quotient field) ofafield F is isomorphic to F.

122. Theorem 4.5（定理）

Theorem 4.5. Let S be a multiplicative subset ofa com1nutative ring R and let T be
any coJnJnutative ring with identity. Iff : R � T is a homon1orphism ofrings such that
f(s) is a unit in T for all s E S, then there exists a unique ho1nomorphism of rings
f : s -IR T such that fq;s = f. The ring s-IR is completely determined (up to iso­
 __,



morphisln) by this property.

123. Corollary 4.6（推论）

Corol lary 4.6. Let R be an integral domain considered as a subring of its quotient
field F. lfE is a field and f : R � E a monomorphism of rings, then there is a unique
monomorphism offields f : F � E such that f I R = f. In particular any field Et con­
 taining R contains an isomorphic copy Ft ofF with R C F1 C E 1 -

124. Theorem 4.7（定理）

Theorem 4.7. Let S be a multiplicative subset of a commutative ring R.
 (i) If I is an ideal in R, then s-11 = { ajs I a e I ; s e s l is an ideal in s-lR.
 (ii) lfJ is another ideal in R, then
 s-1(1 + J) = s-1I + s-1J;


 -
 S-1(1J) = (S-1I)(S-1J) ;
 s-1(1 n J) =s 1I n s-1J.

 REMARKS. S-1 / is called the extension of I in S-1 R. Note that r/s e S-1 / need
not imply that r e I since it is possible to have a/s = rIs with a e /, r t /.

 n




=

125. Theorem 4.8（定理）

Theorem 4.8. Let S be a multiplicative subset of a commutative ring R with identit_v
and let I be an ideal of R. Then s- 11 = s-•R ifand only if s n I � 0.

126. Lemma 4.9（引理）

Lem ma 4.9. Let S be a multiplicative subset of a commutative ring R with identity
and let I be an ideal in R.
 (i) I c (;'s- •(s-•I).
 (ii) Ifl = �Ps-1(J) for some ideal J in s-•R , then S- 11 = J. In other words every
ideal in s-•R is of the form s-•I for some ideal I in R.
 (iii) If P is a prime ideal in R and S n P = 0, then s-tp is a prime ideal in s-•R
and (;'s-1(S-1P) = P.

127. Theorem 4.10（定理）

Theorem 4.10. Let S be a multiplicative subset ofa commurative ring R with identify.
Then there is a one-ro-one correspondence between the set 'U ofprime ideals ofR which
are disjoint from S and the set CV ofprime ideals ofS-1 R, given by P � s-IP.

128. Definition 4.12（定义）

Defi nition 4.12. A local ring is a commutative ring with identity which has a unique
maximal ideal.

 REMARK. Since every ideal in a ring with identity is contained in some maximal
ideal (Theorem 2.1 8), the unique maximal ideal of a local ring R must contain every
ideal of R (except of course R itself).

 EXAMPLE. If p is prime and n > 1 , thenZpn is a local ring with unique maxi­
mal ideaJ (p).


Theorem 4 .. 13. IJ R is a con1mutative ring with identity then the following conditions
are equivalent.
 (i) R is a local ring;
 (ii) all nonunirs ofR are contained in some ideal M #- R ;
 (iii) the nonunits ofR form an ideal.

III.5 Rings of Polynomials and Formal Power Series

129. Theorem 5.1（定理）

Theorem 5.1. Let R be a ring and let R[x] denote the set ofall sequences ofelements
ofR (ao,a t, . . . ) such that ai = 0 for all but a finite number of indices i .
 (i) R[x] is a ring with addition and multiplication defined by:
 (ao,al , . . . ) + (bo,bt , . . .) = (ao + bo,at + bt , . . . )
and
 (ao,at, . . . )(bo,bt , . . .) = (co,Ct , . . . ),
where

 L an-ibi = anbo + an-tbl + . . . + atbn- 1 + aobn = L akbj .
 n
 Cn =

 i=O k +i = n

 (ii) If R is commutative [resp. a ring with identity or a ring with no zero divisors or
an integral domain], then so is R[x].
 (iii) The map R -+
 R[x] given by r � (r ,0,0, . . .) is a monomorphism of rings.

130. Theorem 5.2（定理）

Theorem 5.2. Let R be a ring with identity and denote by x the element (O,lR,O,O, . . . )
of R [ x].
 (i) x n = (0,0, . . . ,0,1R,O, . . .), where 1R is the (n + l)st coordinate.
 (ii) lf r E R , then for each n > 0, r xn = xn r = (0, . . . ,O,r,O, . . . ), where r is the
(n + I )st coordinate.



 (iii) For every nonzero polynomial f in R [x) there exists an integer n e N and ele­
ments a0, , an E R such rhar f = a0x0 + a1 x1 + · · · + anx n . The integer n and
 • • •




elements ai are unique in the sense that f = box0 + btX1 + · · · + bmxm (bi E R) implies
m > n ; ai = bi for i = 1 ,2, . . . , n ; and bi = 0 for n < i < m.

131. Theorem 5.3（定理）

Theorem 5.3. Let R be a ring and denote by R[x1, . . . , xn] the set of all functions
f : Nn � R such that f(u) '# 0 for at m ost a finite number of elements u ofNn.
 {i) R [ x , . . . , Xn] is a ring with addition and multiplication defined by
 (f + g)(u) = f(u) + g(u) and (fg){u) = ,L f(v)g(w),
 v +w -= u
 v,wcN n


where f,g E R [x�, . . . , Xn ] and u E Nn .
 (ii) IfR is commutative [resp. a ring with identity or a ring without zero divisors or
an integral domain ] , then so is R[x1 , . . . , Xn] .
 (iii) The map R � R [ x , . . . , Xn] given by r � fr, where fr(O, . . . , 0) = r and
 .


f(u) = 0 for all other u E Nn , is a monomorphism of rings.

132. Theorem 5.4（定理）

Theorem 5.4. Let R be a ring with identity and n a positive integer. For each
i = 1 ,2, . . . , n let xi E R[x 1 , . . . , X n] be defined by xi(Ei) = 1R and xi(u) = 0 for u '# Ei .
 (i) For each integer k E N, xik(kEi) = lR and X ik(u) = 0 for u '# kEi ;
 (ii) for each (kt, . . . , kn) E Nn ' Xtk1x2k2 • X nkn(ktEl + . + knEn) = 1R and
 •
 •
 . .

X1k1 X2k2 •
 • Xnkn(u) = 0 for U � k1E1 + · · + knEn ;
 •
 ·




 (iii) Xi8Xj t = Xj txis for all s,t E N and all i,j = 1 ,2, . . . , n ;
 (iv) xi tr r x t for all r E R and all t E N;
 = i

 (v) for every polynomial f in R[ x , . . . .. x"] there exist unique elements ak. , . . . 'kn E R,
 ,

indexed by all (k , , . k ) E Nn and nonzero for at most a finite number of (k1 , ,�) E
 . . , " •
 • •




Nn , such that




where the sum is over all (k1, •
 •
 •
 , kJ E N°.

133. Theorem 5.5（定理）

Theorem 5.5. Let R andS be contmutatiL"e rings with identity and cp : R � S a homo­
morphism of rings such that cp(lR) ls . If s . s2 , . . . , Sn E S, then there is a unique
 = ,



homomorphism of rings cp : R[x., . . . , Xn] --+ S such that q; I R = q; and q;(xi) = Si
for i = .
 1 ,2, . . , n. This property completely determines the pofvnomial ring
R[x1 , . . . , Xn] up to isomorphisln.

134. Corollary 5.6（推论）

Corol lary 5.6. If cp : R � S is a homomorphism of commutative rings and
S1,S2, . . . , S11 E S, then the map R [x ., . . . , Xn] � S given by f � cpf(s�, . . . , Sn) is a
homomorphisn1 ofrings.

135. Corollary 5.7（推论）

Corollary 5.7. Let R be a commutativ e ring with identity and n a positive integer.
For each k (1 < k < n) there are isomorphisms ofrings R [x., . . . , xk][x k+ h . . , Xn] ""'
R [x . , . . . , Xn ] f"-1 R[xk+ l· . . . , Xn ][Xt,
 .

 •
 xk].
 • • '

136. Proposition 5.8（命题/性质）

Pro position 5.8. Let R be a ring and denote by Rx the set of all sequences ofele­
ments ofR (ao,at, . . .) .
 (i) R [[ x]J is a ring with addition and n1ultiplication defined by : (ao,at, . . .) +
(bo ,bt , . . ) = (ao + bo,at + bt, . . . ) and (ao,at , . . .)(b0,bt , . . . ) = (co,Ct , . . ), where
 . .



 n n
Cn = L aibn-i = L akbj .
 i=O k +i = n
 (ii) The polynomial ring R[x] is a subring ofRx.
 (iii) If R is commutative [resp. a ring with identity or a ring with no zero divisors or
an integral domain], then so is Rx.

137. Proposition 5.9（命题/性质）

Proposition 5.9. Let R be a ring with identity and f = L aixi � Rx.
 i=O

 {i) f is a unit in Rx if and only if its constant tern1 ao is a unit in R.
 (ii) lfa0 is irreducible in R, then f is irreducible in Rx.

 REMARK. If fe Rx is actually a polynomial with irreducible [resp. unit] con­
stant term then fneed not be irreducible [resp. a unit] in the polynomial ring R[x]
(Exercise 8).

138. Corollary 5.10（推论）

Corollary 5.10. If R is a division ring, then the units in Rx are precisely rnose
power series with nonzero constant ternt. Tire principal ideal (x) consists precisely ofthe
nonunits in R x and is the unique 1naximal ideal of Rx. Thus ifR is afield, Rx is
a local ring.

III.6 Factorization in Polynomial Rings

139. Theorem 6.1（定理）

Theorem 6.1. Let R be a ring and f,g € R[xt, . . . , X n].
 (i) deg(f + g) < max (deg f, deg g).
 (ii) deg(fg) < deg f + deg g.
 (iii) /f R has no zero divisors, deg(fg) = deg f + deg g.
 (iv) Jfn = 1 and the leading coefficient off or g is not a zero divisor in R (in par­
ticular, ifit is a unit), then deg(fg) = deg f + deg g.
 REMARK. The theorem is also true if deg fis taken to mean "degree of fin Xk. n

140. Theorem 6.2（定理）

Theore m 6.2. (The Division Algorithm) Let R be a ring wi(h identity and f,g € R[x]
nonzero polynomials such that the leading coefficient ofg is a unit in R . Then there exist
unique polynomials q,r € R [x] such that
 f = qg + r and deg r < deg g.
 n

141. Corollary 6.3（推论）

Corollary 6.3. (Remainder Theorem) Let R be a ring with identity and
 n

 f(x) = L aixi e R [x] .
 i=O

For any c E R there exists a unique q(x) E R [x] such that f(x) = q(x)(x - c) + f(c).

142. Corollary 6.4（推论）

Corollary 6.4. If F is a field, then the polynomial ring F[x] is a Euclidean domain,
whence F[x] is a principal ideal don1ain and a unique factorization do1nain. The units in
F[x] are precisely the nonzero constant polynomials.

143. Definition 6.5（定义）

Defin ition 6.5. Let R be a subring of a commutative ring S, c 1 ,c2 , . . . , Cn c: S and

f = L aix� . x!in e; R [x�, . . . , X n] a polynomial such tha t f(ct,C2, . . . Cn) = 0.
 m

 il
 0 o



 i=O
Then (Ct,C2, , C n) is said to be a root or zero of f (or a solution of the polynomial
 • • •

144. Theorem 6.6（定理）

Theore m 6.6. Let R be a commutative ring with identity and f e; R [x]. Then c e; R is a
root of f if and only ifx - c divides f.

145. Theorem 6.7（定理）

Theore m 6.7. If D is an integral domain contained in an integral domain E and
f e; D [x] has degree n, then f has at most n distinct roots in E .

146. Proposition 6.8（命题/性质）

Pro position 6.8. Let D be a unique factorization domain with quotient field F and let
 n


f = L aixi E D[x]. lf u = cjd E F with c and d relatively prime, a11.d u is a root of f,
 i=O
then c divides ao and d divides a n .

 4Commutativity is not essential in the definition provided one distinguishes ''left roots''
and "right roots" {the latter occur when f is written f L x�·· · · · x!''�ai).=

147. Lemma 6.9（引理）

Lem ma 6.9. Let D be an integral domain andf = L: aixi E D[x]. Let f' E D [x] be the
 n

polynomial f' = L kakxk-1 = a l + 2a2x + 3a3x2 + . . . + nanxn-1• Then for all
 i=O


 k=l
f,g e D [x] and c e D :
 (i) (cf)' = cf' ;
 (ii) (f + g)' = f' + g' ;
 (iii) (fg) ' = f'g + fg' ;
 (iv) (gn)' = ngn-lg'.

148. Theorem 6.10（定理）

Theorem 6.10. Let D be an integral domain which is a subring oj·an integral domain
E . Let f e D [x) and c e E .
 (i) c is a multiple root off if and only if f(c) = 0 and f'(c) = 0.
 (ii) lfD is afield and f is relatively prime to f', then f has no multiple roots in E.
 (iii) IJD is a field, f is irreducible in D [x] and E contains a root off, then f has no
ntultip/e roots in E if and only iff' '¢ 0 .

149. Lemma 6.11（引理）

Lem ma 6.11. (Gauss) If D is a unique factorization domain and f,g e D[x], then
C(fg) = C{f)C(g). In particular, the product of primitive polynomials is primitive.

150. Lemma 6.12（引理）

Lemma 6.12. Let D be a unique factorization domain with quotient field F and let f
and g be primitive polynomials in D[x]. Then f and g are associates in D[x] ifand only if
they are associates in F[x].

151. Lemma 6.13（引理）

Lem ma 6.13. Let D be a unique factorization domain with quotient field F and f a
primitive polynomial ofpositive degree in D[x] . Then f is irreducible in D[x] ifand only
iff is irreducible in F[x].

152. Theorem 6.14（定理）

Theore m 6.14. IJD is a unique factorization domain, then so is the polynomial ring
D[x., . . , Xn ] .
 .




 REMARK. Since a field F is trivially a unique factorization domain, F[xh . . . , Xn]
is a unique factorization domain.

153. Theorem 6.15（定理）

Theorem 6. 15. (Eisenstein's Criterion). Let D be a unique factorization domain with
 n

quotient field F. Iff = L aixi E D [x] deg f > I and p is an irreducible element ofD
 ,

 i=O
such that
 p .f'a n ; p f ai for i = 0, 1 , . . . • n - 1 ; P2 1' ao,



then f is irreducible in F[x]. Iff is primitive., then f is irreducible in D[x].

Chapter IV: Modules

IV.1 Modules, Homomorphisms and Exact Sequences

154. Definition 1.1（定义）

Defin ition 1.1. Let R be a ring. A (left) R-module is an addiriDe abelian group A to­
gether with a function R X A --t A (rhe image of (r ,a) being denoted by ra) such that
for all r,s c R and a, b € A :
 (i) r(a + b) = ra + r b.
 (ii) (r + s)a = ra + sa.
 (iii) r(sa) = (rs)a.
lfR has an identity element 1 R and
 (iv) 1 Ra = a for all a € A,
then A is said to be a unitary R-module. /fR is a division ring, then a unitary R-module
is called a (left) vector space.

 A (unitary) right R-module is defined similarly via a function A X R -+ A de­
noted (a,r) � ar and satisfying the obvious analogues of (i)-(iv). From now on, un­
less specified otherwise, "R-module" means uleft R-module.. and it is understood
that all theorems about left R-modules also hold, mutatis mutandis, for right R­
modules.
 A given group A may have many different R-module structures (both left and
right). If R is commutative, it is easy to verify that every left R-module A can be given
the structure of a right R-module by defining ar = ra for r e R, a € A (commutativity
is needed for (iii) ; for a generalization of this idea to arbitrary rings, see Exercise 1 6).
Unless specified otherwise, every module A over a commutative ring R is assumed to
be both a left and a right module with ar = ra for all r E R, a € A.
 If A i s a module with additive identity element OA over a ring R with additive
identity OR, then it is easy to show that for all r € R, a e A :



In the sequel OA,OR,O € Z and the trivial module { 0 l will all be denoted 0.
 It also is easy to verify that for all r E R, n € Z and a € A :

 ( -r)a = - (ra) = r( -a) and n(ra) = r(na),

where na has its usual meaning for groups (Definition 1.1 .8, additive notation).


 EXAMPLE. Every additive abelian group G is a unitary Z-module, with
na (n c Z, a € G) given by Definition 1. 1 .8.

 EXAMPLE. If S is a ring and R is a subring, then S is an R-module (but not
vice versa !) with ra (r € R , a € S) being multiplication in S. In particu1ar, the rings
R[xh . . . , Xm] and Rx are R-rnodules.



 EXAMPLES. If I is a lefT ideal of a ring R, then I is a left R-module with
ra (r E R,a E /) being the ordinary product in R. In particular, 0 and R are R-modules.
Furthermore, since I is an additive subgroup of R, Rj I is an (abelian) group. R/ I is
an R-module with r(r t + I) = rr1 + I. R/ I need not be a ring, however, unless I is a
two-sided ideal.

 EXAMPLE. Let R and S be rings and <P : R � S a ring homomorphism. Then
every S-module A can be made into an R-module by defining rx (x E A) to be <P(r)x.
One says that the R-module structure of A is given by pullback along <P ·


 EXAMPLE. Let A be an abelian group and End A its endomorphism ring (see
p. 1 1 6). Then A is a unitary (End A)-module, with fa defined to be f(a) (for a E A,
/e End A).

 '

 EXAMPLE. If R is a ring, every abelian group can be made into an R-module
with trivial module structure by defining ra = 0 for all r E R and a e A .

155. Definition 1.2（定义）

Defi n ition 1.2. Let A and B be modules over a ring R . A function f : A � B is an
R-module homomorphism provided that for all a,c E A and r e R :

 f(a + c) = f(a) + f(c) and f(ra) = rf(a).
If R is a division ring, then an R-module homomorphism is called a linear trans­
formation.

 When the context is clear R-module homomorphisms are called simply homo­
morphisms. Observe that an R-module homomorphism· f : A ----+ B is necessarily a
homomorphism of additive abelian groups . Consequently the same terminology is
used: / is an R-module monomorphism [resp. epimorphism , isomorphism] if it is in­
jective [resp. surjective, bijective] as a map of sets. The kernel of .fis its kernel as a
homomorphism of abelian groups, namely Ker f = { � E A I f(a) = 0 } . Similarly
the image of /is the set Im f = { b E B l b = f(a) for some a E A } . Finally, Theorem
1.2.3 implies :
 (i) f is an R-module monomorphism if and only if Ker f = 0;
 (ii) f : A ----+ 8 is an R-module isomorphism if and only if there is an R-module
homomorphism g : B � A such that gf = L -1 and fg = 1 u .

 EXAMPLES. For any modules the zero map 0 : A - � B given by a J--, 0 (a E A) is
a module homomorphism. Every homomorphism of abelian groups is a Z-module
homomorphism. If R is a ring, the map R[x] ----+ R[x] given by f � xf(for example,
(x2 + 1 ) � x(x2 + 1 )) is an R-module homomorphism, but not a ring homo­
morphism.

 REMARK. For a given ring R the class of all R-modules [resp . unitary
R-modules] and R-module homomorphisms clearly forms a (concrete) category. I n
fact, one can define epimorphisms and monomorphisms strictly i n categorical terms
(objects and morphisms only - no elements) ; see Exercise 2 .

156. Definition 1.3（定义）

Defin ition 1.3. Let R be a ring, A an R-module and B a nonempty subset of A. B is a
submodule of A provided that B is an additive subgroup of A and rb c B for all r E R,
b c B. A submodule of a vector space over a division ring is called a subspace.

 Note that a submodule is itself a module. Also a submodule of a unitary module
over a ring with identity is necessarily unitary.

 EXAMPLES. If R is a ring and f: A � B is an R-module homomorphism, then
Ker fis a submodule of A and Im fis a submodule of B. If C is any submodule of B,
then f-1( C) = { a c A I f(a) E C} is a submodule of A .

 EXAMPLE. Let I be a left ideal of the ring R, A an R-module and S a nonempty

subset of A . Then IS = { t;1 }
 r;a; I r; e I; a; e S; n e N* is a submodule of A (Exer­
cise 3 ). Similarly if a e A , then Ia = { ra I r E /} is a submodule of A .


 iel
 EXAMPLE. If { Bi I i E /} is a family of submodules of a module A, then n Bi is

easily seen to be a submodule of A .

157. Definition 1.4（定义）

Defin ition 1.4. IfX is a subset ofa module A over a ring R , then the intersection of
all submodules of A containing X is called the submodule generated by X (or spanned
by X).

 If X is finite, and X generates the module B, B is said to be finitely generated. If
X = 0, then X clearly generates the zero module. If X consists of a single element,
X = { a } , then the submodule generated by X is called the cyclic (sub)module gen­
erated by a. Finally, if { Bi I i E /} is a family of submodules of A , then the submodule

 iel
generated by X = U Bi is called the sum of the modules Bi. If the index set I is finite,
the sum of Bt, . . . , Bn is denoted Bt + B2 + · · · + Bn.

158. Theorem 1.5（定理）

Theorem 1.5. Let R be a ring, A an R-module, X a subset ofA , { Bi I i E I } a family
ofsubmodules of A and a c A . Let R a = { ra I r E R } .
 (i) Ra is a submodule of A and the n1ap R � R a given by r � ra is an R-module
epimorphism.
 (ii) The cyclic submodule C generated by a is { ra + na J r c R; n E Z} . lfR has an
identity and C is unitary, then C = Ra.
 (iii) The submodule D generated by X is

 I t r;a; + t n;b; I s,t e N*; a;,b; e X ;r; e R ; n; E
 l i= l j= l
 z}.
lfR has an identity and A is unitary, then

 D = RX = {tt=l
 }
 ri ai I s E N*; ai c X ; ri E R .

 r
 (iv) The sum of the family { Bi I i e I } consists ofallfinite sums bi1 + · · · + bin with
bik e Bi k ·

159. Theorem 1.6（定理）

Theorem 1.6. Let B be a submodule ofa module A over a ring R . Then the quotient
group A/B is an R-module wirh the action ofR on A/B given by:
 r(a + B) = ra + B for all r e R,a e A.

The map 1r : A --. A/B given by a � a + B is an R-modu/e epimorphism with kernel B.

 The map 1r is called the canonical epimorphism (or projection).

160. Theorem 1.7（定理）

Theorem 1.7. If R is a ring and f : A --. B is an R-module homomorphism and C is a
ti:ubmodule ofKer f, then there is a unique R-module homomorphism f : A/C � B such
that f (a + C) = f(a) for all a e A ; bn f = lm f and Ker f = Ker f/C. f is an R-module
isomorphism ifand only iff is an R-module epimorphism and � = Ker f. In particular,
A/Ker f ,.-....; lm f.

161. Corollary 1.8（推论）

Corol lary 1.8. /fR is a ring and A' is a submodule of the R-module A and B' a sub­
module of the R-module B and f : A � B is an R-module homomorphism such that
f(A') C B', rhen f induces an R-module homomorphism f : A/ A' --. B/B' given by
a + A' � f(a) + B'. f is an R-module isomorphism if and only iflm f + B' = B and
f 1( B') C A'. ln particular iff is an epimorphism such that f(A') = B' and Ker f C A',
-




then f is an R-module isomorphism.

162. Theorem 1.9（定理）

Theorem 1.9. Let B and C be submodules of a module A over a ring R.
 (i) There is an R-module isomorphism B/(B n C) (B + C)jC ;
 r-...;



 (ii) ifC C B, then B/C is a submodule of AjC, and rhere is an R-module isomor­
phism (A/C)/(B/C) A/B. r-...;

163. Theorem 1.10（定理）

Theo rem 1. 10. If R is a ring and B is a submodule ofan R-module A, then there is a
one-to-one correspondence between the set ofall submodules ofA containing B and the
set ofall submodules ofA/B, given by C � C/B. Hence every submodule of A/B is of
the form CjB, where C is a submodule of A which contains B.

164. Theorem 1.11（定理）

Theorem 1.11. Let R be a ring and I A1 I i e I } a nonempty family of R-modules,
II Ai the direct product of the abelian groups A;, and I: Ai the direct sum of the
iel iel
abelian groups Ai.
 (i) II Ai is an R-module wirh the action ofR given by r I ai } - { rai I .
 (ii) I: Ai is a submodule of II A i.
 iei


 �I �I
 (iii) For each k E I, the canonical projection 7rk : II Ai � Ak (Theorem 1.8. 1) is
an R-module epimorphism.
 (iv) For each k E I, the canonical injection tk : Ak � I: Ai (Theorem 1.8.4) is an
R-module monomorphism.

165. Theorem 1.13（定理）

Theorem 1.13. lf R is a ring, { Ai l i e I } a fami/y of R-modu/es, D an R-module, and
{ �i : Ai � D I i e I } a family of R-modu/e homomorphisms, then there is a unique
R-module homomorphism V; : L Ai � D such that � Li = t/;i for all i e I. L Ai is
uniquely determined up to isomorphism by this property. ln other words, L Ai is a co­
 iel iel


 iei
product in the category of R-modules.

166. Theorem 1.14（定理）

Theorem 1.14. Ler R be a ring and A,A.,A2, , An R-modules. Then A t"... A 1 EB
 • • •




A2 EB · · · EB An if and only iffor each i = 1 ,2, . . , n rhere are R-module homomor­
 .



phisms 1ri : A � Ai and Li : Ai � A such that
 (i) 1riLi = J A i for i = 1 ,2, . . . , n ;
 (ii) 7rj L i = 0 for i ¢ j ;
 (iii) LI7rt + L2'1r2 + · · · + Ln7rn = l A .

167. Theorem 1.15（定理）

Theorem 1.15. Let R be a ring and { Ai I i E I } a family ofsubmodu/es ofan R-module
A such that

 (i) A is the sum of the family { Ai I i E I } ;
 (ii) for each k E I, Ak n Ak * = 0, where Ak * is the sum ofthe family { A i I i � k } .
Then there is an isomorphism A "" L Ai.
 iel

168. Definition 1.16（定义）

Defin ition 1. 16. A pair ofmodule homomorphisms, A � B � C, is said to be exact
at B provided lm f = Ker g. A finite sequence ofmodule homomorphisms, Ao � At �
A2 � - · · �� An-1 � An, is exact provided lm fi = Ker fi + t for i = 1 ,2, . . . , n - 1 . An

tn .
 . J'nzte sequence of mouu,e . fi-1 fi fiTl fi+l
 ,I 1 homomorph tsms, · · · � Ai- 1 � Ai ------+- A i+l � •
 .
 • •
 , .{;
 IS exact

provided lnz fi Ker fi+l for all i E Z.
 =




 When convenient we shall abuse the language slightly and refer to an exact se­
quence of modules rather than an exact sequence of module homomorphisms.

 EXAMPLES. Note first that for any module A, there are unique module homo­
morphisms 0 -4 A and A -4 0. If A and B are any modules then the sequences
0 � A � A E8 B � B � 0 and 0 � B � A E8 B � A � 0 are exact, where the t's
and 1r's are the canonical injections and pro�ections respectively. Similarly, if C is a
submodule of D, then the sequence 0 � C � D � D/C � 0 is exact, where i is the

 r

morphism, then A/Ker /[resp. B/Im f] is called the coimage of f [resp. cokernel off]
and denoted Coim f [resp. Coker f] . Each of the following sequences is exact :
A � B � Coker f 0, where the unlabeled maps are the obvious inclusions and
 --4

projections.

 REMARKS. 0 � A � B is an exact sequence of module homomorphisms if and
only if f is a module monomorphism. Similarly, B !!.._, C --4 0 is exact if and only if g
is a module epimorphism. If A !:. B � C is exact, then gf = 0. Finally if A � B �
C --4 0 is exact, then Coker f = B/Im f = BjKer g = Coim g C. An exact se- "'-'



quence of the form 0 A � B .!!....:. C � 0 is called a short exact sequence ; note that f
 --4

is a monomorphism and g an epimorphism. The preceding remarks show that a short
exact sequence is just another way of presenting a submodule (A "'-' Im f) and its
quotient module (B/Im f = B/Ker g "'-' C).

169. Lemma 1.17（引理）

Lemma 1.17. (The Short Five Lemma) Let R be a ring and




a commutative diagran1 of R-modules and R-modu/e homomorphisms such that each
row is a short exact sequence. Then
 (i) a,')' monomorphisms ::::::} {3 is a monomorphism;
 (ii) a,')' epimorphisms ::::::} {3 is an epimorphism;
 (iii) a,')' isomorphisms => {3 is an isomorphism.

170. Theorem 1.18（定理）

Theorem 1.18. Let R be a ring and O � AI ---7 B ---4 A2 � 0 a short exact sequence of
R-module homomorphisms. Then the following conditions are equivalent.
 (i) There is an R-module homomorphism h : A2 ---4 B with gh = 1 A2 ;
 (ii) There is an R-module homomorphism k : B ---7 A1 with kf = 1 A t ;
 (iii) the given sequence is isomorphic (with identity maps on A1 and A;) to the
direct sum short exact sequence 0 ---4 A. � At EB A2 � A2 ---7 0 ; in particular
B r-..J A I EB A2.

 A short exact sequence that satisfies the equivalent conditions of Theorem 1 .1 8 is
said to be split or a split exact sequence.

IV.2 Free Modules and Vector Spaces

171. Theorem 2.1（定理）

Theorem 2.1. Let R be a ring with identity. The following conditions on a unitary
R-module F are equivalent:
 (i) F has a nonempty basis;
 (ii) F is the internal direct sum of a family ofcyclic R-modules, each of which is
isomorphic as a left R-module to R ;
 (iii) F is R-modu/e isomorphic to a direct sum of copies of the left R-module R ;
 (iv) there exists a nonempty set X and a function L : X ---+ F with the following
property: given any unitary R-module A and function f : X A, there exists a unique
 ---+


R-module homomorphism f : F � A such that f L = f. In other words, F is a free object
in the category of unitary R-modules.

 The theorem is proved below. A unitary module F over a ring R with identity,
which satisfies the equivalent conditions of Theorem 2 . 1 , is called a free R-module on
the set X. By Theorem 2. 1 (iv), F is a free object in the category of all unitary left
R-modules. But such an F is not a free object in the category of all left R-modules
(Exercise 1 5). By definition the zero module is the free module on the empty set.
 It is possible to define free modules in the category of all left R-modules over an
arbitrary ring R (possibly without identity) ; see Exercise 2. Such a free module is not
isomorphic to a direct sum of copies of R, even when R does have an identity (Exer­
cise 2). In a few carefully noted inst�nces below, certain results are also valid for
these free modules in the category of all left R-modules. However, unless stated
otherwise, the term "free module ... will always mean a unitary free module in the
sense of Theorem 2. 1 .

172. Corollary 2.2（推论）

Corollary 2.2. Every (unitar.v) module A over a ring R (with identity) is the /unnomor­
phic image ofa free R-module F. IfA is finitely generated, then F 1nay be chosen to be
finitely generated.

 REMARK . Corollary 2.2 and its proof are valid i f the words in parentheses are
deleted and "free module" is taken to mean a free module in the category of a ll left
modules over an arbitrary ring (as defined in Exercise 2).

 l

173. Lemma 2.3（引理）

Lem ma 2.3. A maxima/ linearly independent subset X of a vector space V over a
division ring D is a basis of V.

174. Theorem 2.4（定理）

Theorem 2.4. Every vector space V over a division ring D has a basis and is therefore
a free D-module. More generally every linearly independent subset of V is contained in
a basis of V.

 The converse of Theorem 2.4 is also true, namely, if every unitary module over a
ring D with identity is free, then D is a division ring (Exercise 3 . 1 4).

175. Theorem 2.5（定理）

Theorem 2.5. If V is a vector space ocer a division ring D and X is a subset that
spans V, then X contains a basis ofV.

 r

176. Theorem 2.6（定理）

Theorem 2.6. Let R be a ring with identity and F a free R-module with an infinite
basis X . Then every basis ofF has the same cardinality as X.

177. Theorem 2.7（定理）

Theorem 2.7. If V is a vector space over a division ring D, then any two bases of V
have the same cardinality.

178. Definition 2.8（定义）

Defin ition 2.8. Let R be a ring with identity such that for every free R-module F, any
two bases of F haoe the same cardinality. Then R is said to have the invariant dimension
property and the cardinal number ofany basis of F is called the dimension (or rank) of
F over R.

 Theorem 2.7 states that every division ring has the invariant dimension property.
We shall follow the widespread (but not universal) practice of using "dimension"'
when referring to vector spaces over a division ring and "rank" when referring to free
modules over other rings. The dimension of a vector space V over a division ring D
will be denoted here by dimnV. The properties of dimnV will be investigated after
Corollary 2 . 1 2 . Results 2.9-2. 1 2 are not needed in the sequel, except in Sections
IV.6 and Vll.5.

179. Proposition 2.9（命题/性质）

Proposition 2.9. Let E and F be free modules over a ring R that has the inoariant
dimension property. Then E ""' F ifand only ifE and F hat,e the same rank.

180. Lemma 2.10（引理）

Lem ma 2.10. Let R be a ring with identity, I (-:;C R) an ideal ofR, F a free R-module
with basis X and 7r : F � F/ I F the canonical epimorphism. Then F/IF is a free R/I­
module with basis 7r(X) and 17r(X )I = l X I .




 Recall that IF = { t r;a; r; a, n EN*}
 ,=1
 I c /, c F, and that the action of RlI on
FlIF is given by (r + l)(a + IF) = ra + IF (Exercise 1 .3).
 n

181. Proposition 2.11（命题/性质）

Proposition 2.11. Let f : R � S be a nonzero epimorphism ofrings with identity. If
S has the invariant dimension property, then so does R.

182. Corollary 2.12（推论）

Corollary 2.12. lfR is a ring with identity that has a homomorphic image which is a
division ring, then R has the invariant dimension property. In particular, every com­
mutative ring with identity has the invariant dimension property.

183. Theorem 2.13（定理）

Theorem 2.13. Let W be a subspace of a l'ector space V over a division ring D.

 (i) dimoW < dimoV;
 (ii) ifdimoW dimoV and dimoV is finite, then W = V;
 =



 (iii) dimoV = dimoW + dimo(VIW).

184. Corollary 2.14（推论）

Corollary 2.14. Iff : V ----.. V' is a linear transformation of vector spaces over a divi­
sion ring D, then there exists a basis X of V such that X n Ker f is a basis ofKer f and
{ f(x) I f(x) � 0, x e X l is a basis of lm f. In particular,
 dimoV = dimo(Ker f) + dimo(lm f).

185. Corollary 2.15（推论）

Corollary 2. 15. lf V and W are fin ite dimensional subspaces ofa cector space over a
division ring D, then
 dimoV + dimoW = dimo(V n W) + dimo( V + W) .

186. Theorem 2.16（定理）

Theorem 2.16. Let R,S,T be division rings such that R C S C T. Then
 dimRT = (dimsT)(dimnS).
Furthermore, dimnT is finite if and only ifdim sT and dim RS are finite.

IV.3 Projective and Injective Modules

187. Definition 3.1（定义）

Defin ition 3.1. A module P over a ring R is said tv be projective ifgiven any diagram
ofR-module homo1norphisn1S



with bottom row exact (that is, g an epimorphism), there exists an R-module homo­
ntorphism h : P � A such that the diagram




is commutative (that is, gh = f).

 The theorems below will provide several examples of projective modules. We
note first that if R has an identity and P is unitary, then P is projective if and only if
for every pair of unitary modules A, B and diagram of R-module homomorphisms




with g an epimorphism, there exists a homomorphism h : P � A with gh = f. For
= RB2. Exercise 1 . 1 7 shows further that f(P) C Bt and g ! At is an epimorphism
A 1 Bt. so that we have a diagram of unitary modules :
 �




Thus the existence of h : P � A with gh = f is equivalent to the existence of
h : P � At with gh = f.

188. Theorem 3.2（定理）

Theorem 3.2. Every free module F over a ring R with identity is projective.

 REMARK. The Theorem is true if the words "with identity'' are deleted and F is
a free module in the category of all left R-modules (as defined in Exercise 2.2). The

189. Corollary 3.3（推论）

Corol lary 3.3. EDery module A over a ring R is the homomorphic image ofa projec­
tive R-modu/e.

190. Theorem 3.4（定理）

Theorem 3.4. Let R be a ring. The following conditions on an R-module P are
equivalent.
 (i) P is projective;
 (ii) every short exact sequence 0 --+ A � B � P ---+ 0 is split exact (hence
B A EB P);
 '"'-'



 (iii) there is a free module F and an R-module K such that F � K E8 P.

 REMARK. The words "free module" in condition (iii) may be interpreted in
the sense of Theorem 2.1 if R has an identity and P is unitary, and in the sense of
Exercise 2.2 otherwise. The proof is the same in either case.

191. Proposition 3.5（命题/性质）

Pro position 3.5. Let R be a ring. A direct sum ofR-modules L: Pi is projective if.
 ial
and only if each Pi is projective.

192. Definition 3.6（定义）

Defin ition 3.6. A module J over a ring R is said to be injective ifgiven any diagram
ofR-module homomorphisms





with top row exact (that is, g a monomorphism ), there exists an R-1nodule ho1no1nor­
phism h : B --+ J such that the diagram




is commutative (that is, hg = f).

 Remarks analogous to those in the paragraph following Definition 3.1 apply here
to unitary injective modules over a ring with identity. It is not surprising that the
duals of many (but not all) of the preceding propositions may be readily proved. For
example since in a category products are the dual concept of coproducts (direct
sums), the dua! of Proposition 3.5 is

193. Proposition 3.7（命题/性质）

Proposition 3.7. A direct product of R-modules II Ji is injective ifand only ifJi is
 iel
injective for every i e I .

194. Lemma 3.8（引理）

Lem ma 3.8. Let R be a ring with identity. A unitary R-module J is injective if and
only iffor every left ideal L ofR, any R-module homomorphism L --+ J may be ex­
tended to an R-module homomorphism R --+ J.

195. Lemma 3.9（引理）

Lem ma 3.9. An abelian group D is divisible if and only if D is an injective (unitary)
Z-module.

196. Lemma 3.10（引理）

Lem ma 3.10. Every abelian group A may be embedded in a divisible abelian group.

197. Lemma 3.11（引理）

Lem ma 3.11. If J is a divisible abelian group and R is a ring with identity, then
Homz(R,J) is an injective left R-module.

198. Proposition 3.12（命题/性质）

Proposition 3.12. Every unitary module A over a ring R with identity may be em­
bedded in an injective R-modu/e.

199. Proposition 3.13（命题/性质）

Pro position 3.13. Let R be a ring with identity. The following conditions on a
unitary R-module J are equivalent.
 (i) J is injective;
 (ii) every short exact sequence 0 � J � B � C � 0 is split exact (hence
B � J Ef) C);
 (iii) J is a direct summand ofany module B of which it is a submodu/e .

IV.4 Hom and Duality

200. Theorem 4.1（定理）

Theorem 4.1. Let A,B,C,D be modules over a ring R and cp : C � A andl/; : B � D
R-module homomorphisms. Then the map () : HomR(A,B) --+ HomR(C,D) given by
f f-.-+ 1/;fcp is a homomorphism of abelian groups.

201. Theorem 4.2（定理）

Theorem 4.2. Let R be a ring. 0 � A � B � C is an exact sequence ofR-modules if
and only iffor every R-module D
 0 � HomR(D,A) � Homn(D,B) � Homn(D,C)
is an exact sequence of abelian groups.

202. Proposition 4.3（命题/性质）

Proposition 4.3. Let R be a ring. A � B � C � 0 is an exact sequence of R-mod­
ules if and only iffor every R-module D
 0 -t Hom R(C, D) � HomR(B,D) � HomR(A,D)
is an exact sequence of abelian groups.

203. Proposition 4.4（命题/性质）

Pro position 4.4. The following conditions on modules over a ring R are equivalent.

 (i) 0 ---+ A � B � C ---+ 0 is a split exact sequence ofR-modules;
 (ii) 0 ---+ HomR(D,A) � HomR(D,B) � HomR(D,C) ---+ 0 1s a split exact se­
quence ofabelian groups for every R-module D;
 (iii) 0 ---+ HomR(C,D) � HomR(B,D) � HomR(A,D) ---+ 0 is a split exact se­
tjUence of abelian groups for every R-module D.

204. Theorem 4.5（定理）

Theorem 4.5. The following conditions on a module P over a ring R are equivalent
 (i) P is projective;
 (ii) ift/.t : B ---+ C is any R-module epimorphism then � : HomR(P,B) ---+ HomR(P,C)
is an epimorphism of abelian groups;
 (iii) if 0 ---+ A :!� B � C ---+ 0 is any short exact sequence of R-modules, then
0 ---+ HomR(P,A) � HomR(P,B) � HomR(P,C) ---+ 0 is an exact sequence of abelian
groups.

205. Proposition 4.6（命题/性质）

Pro position 4.6. The following conditions on a module J over a ring R are equivalent.
 (i) J is injective; _




 (ii) ifO :A � B is any R.-module monomorphism, then O :HomR(B,J) � HomR(A,J)
is an epimorphism of abelian groups;
 (iii) if 0 � A � B -S C � 0 is any short exact sequence of R-modules, then
0 � HomR(C,J) � HomR(B,J) � HomR(A,J) � 0 is an exact sequence of abelian
groups.

206. Theorem 4.7（定理）

Theorem 4. 7. Let A,B, l Ai I i E I } and { Bj I j E J } be modules over a ring R . Then
there are isomorphisms ofabelian groups:
 (i) HomR('£ AhB ) "' II HomR(Ai,B);
 iel iel
 (ii) HomR(A, II Bj) � II HomR(A,Bj).
 jeJ jeJ


 REMARKS. If I and J are finite, then L Ai = II Ai and L B1 = II Bi. If I
 iel iel jeJ jeJ
and J are infinite, however, the theorem may be false if the direct product II is re­
placed by the direct sum L (see Exercise 1 0).

207. Theorem 4.8（定理）

Theorem 4.8. Let R and S be rings and let RA, RBs, RCs, RD be (bi)modules as in­
dicated.
 (i) HomR(A,B) is a right S-module, with the action ofS gicen by (fs)(a) = (f(a))s
 ·



(s e S; a e A ; f e HomR(A,B)) .
 (ii) If 'fJ : A ___. A' is a homomorphism of left R-n1odu/es, then the induced map
<P : HomR(A ' ,B ) � HomR(A,B ) is a homomorphism of right S-modules.
 (iii) HomR(C ,D) is a left S-module, with the action of S given by (sg)(c) = g(cs)
(s e S ; c e C; g c: Hon1R(C,D)) .
 (iv) lfl/1 : D � D' is a homomorphism of/eft R-modules, then � : HomR(C,D) ___.
HomR(C,D') is a homomorphisn1 of/eft S-modu/es.

208. Theorem 4.9（定理）

Theorem 4.9. If A is a unitary left module over a ring R with identity then there is
an isomorphism ofleft R-modules A 1"'../ HomR(R,A) .

209. Theorem 4.10（定理）

Theorem 4.10. Let A,B and C be left modules over a ring R .
 (i) If <P : A C is a hon101norphism of left R-modules, then the induced1nap
 ---4

({) : C* = HomR(C,R) HomR(A, R ) A* is a hon1omorphism ofriglu R-modules.
 � =



 (ii) There is an R-module iso1norphisn1 (A E8 C)* '""' A* EB C*.
 (iii) /fR is a division ring and 0 A � B -S C 0 is a short exact sequence of
 � �



left vector SFaces, then 0 C* £ B * � A * ----) 0 is a short exact sequence of right
 �


veetor spaces.

210. Theorem 4.11（定理）

Theorem 4.11. Let F be a free left module ocer a ring R with identity. Let X be a
basis of F and for each x e X let fx : F � R be given by fx(Y) = !Jxy (y e X). Then
 (i) { fx I x e X l is a linearly independent subset ofF* .of cardinality lXI ;
 (ii) ifX is finite, then F * is a free right R-n1odule with basis { fx I x e X } .

 REMARKS. The homomorphisms fx are well defined since F is free with basis X
(Theorem 2.1). In part (ii), { fx I x e X } is called the dual basis to X. This theorem is
clearly true for any vector space V over a division ring by Theorem 2.4. In particular,
if V is finite dimensional, then Proposition 2.9 and Theorem 4.1 1 imply that dim V
 = dim V* and V V*. However, if V is infinite dimensional then dim V* > dim V
 rov



(Exercise 1 2). More generalJy, if F is a free module over an arbitrary ring (for ex­
ample, Z), F* need not be free (see Exercise 1 0).

211. Theorem 4.12（定理）

Theorem 4.12. Ler A be a left module over a ring R.
 (i) There is an R-module homomorphism () : A ---+ A**.
 (ii) /fR has an identity and A is free, then () is a monomorphism.
 (iii) lfR has an identity and A is free with a finite basis, then () is an isomorphism.

 A module A such that () : A � A** is an isomorphism is said to be reflexive.

IV.5 Tensor Products

212. Definition 5.1（定义）

Defin ition 5.1. Let A be a right module and B a left n1odule ocer a ring R. Let F be
the free abelian group on the set A X B. Let K be the subgroup ofF generated by all
elements of the following forms ( for all a ,a' E A ; b,b' E B ; r E R) :
 (i) (a + a ' ,b) - (a,b) - (a',b) ;
 (ii) (a ,b + b') - (a,b) - (a,b') ;
 (iii) (ar,b) - (a,rb).

The quotient group F/K is called the tensor product of A and B ; it is denoted A @R B
(or simply A (8) B ifR = Z). The coset (a,b) + K ofthe el�ment (a, b) in F is denoted
a (8) b ; the coset of(O,O) is denoted 0.


 Since F is generated by the set A X B, the quotient group F/ K = A @R B is
generated by all elements (cosets) of the form a (8) b (a E A, b E B). But it is not true
that every element of A @R B is of the form a @ b (Exercise 4). F or the typical ele-
 r

ment of F is a sum L ni(ai,bi) (ni E Z, ai E A, bi E B) and hence its coset i n A @R B
 i=l
 r

= F/K is of the form L niCai (8) b1). Furthermore, since it is possible to choose
 i= l
different representatives for a coset, one may have a ® b a' @ b' in A @R B, but
 =



a � a' and b � b' ( Exercise 4). It is a lso possible to have A @R B = 0 even though
A � 0 and B � 0 (Exercise 3).

 Definition 5.1 implies that the generators a (8) b of A @n B satisfy the follow­
ing relations (for a l l a,ai E A , h,hi E B, and r E R) :

 (a1 + a2) (8) b = a1 (8) b + a2 (8) b; (6)
 a (8) (b1 + b2) = a (8) b1 + a (8) b2; (7)
 ar (8) b = a (8) rb. (8)
The proof of these facts is straightforward ; for example, since (at + a2 h) - (a�,b) -
 .,



(a2,b) E K, the uzero coset,'" we have
 [(a t + a2,b) + K] - ((a1 ,h) + K] - ((a2,b) + K] = K ;
or in the notation (a,b) + K = a (8) b,

 (a1 + a2) (8) b - a1 (8) b - a2 (8) b = 0.
Indeed an alternate definition of A @R B is that it is the abelian group with genera­
tors all symbols a (8) b (a E A, h E B), subject to the relations (6)-(8) above. Further­
more, since 0 is the only element of a group satisfying x + x = x, it is easy to see
that for a l l a E A, b E B:

 a @ 0 = 0 @ b = 0 @ 0 = 0.



 Given modules AR a nd J3 over a ring R, it is easy to verify that the map
i : A X B -4 A @R B given by (a,b) � a @ b is a middle linear map . The map i is
called the canonical middle linear map. Its importance is seen in

213. Theorem 5.2（定理）

Theorem 5.2. Let AR and RB be modules over a ring R, and let C be an abelian group.
lf g : A X B � C is a middle linear n1ap, then there exists a unique group homomor­
phisn1 g : A @R B -4 C such that gi = g, where i : A X B --} A @R B is the canonical
middle linear map. A @R B is uniquely detern1ined up to isomorphism by this property.
In other words i : A X B -4 A @R B is universal in the category mt(A,B) ofall middle
linear maps on A X B.

214. Corollary 5.3（推论）

Corollary 5.3. If AR, AR', nB and RB' are modules over a ring R and f : A � A',
g : B __, B' are R-module homon1orphisms, then there is a unique group homomorphism
A @R B -4 A' @R B' such that a @ b � f(a ) @ g(b) for all a E A, b E B.

215. Proposition 5.4（命题/性质）

Pro po sition 5.4. IfA !.... B � C ---+ 0 is an exact sequence of/eft modules over a ring
R and D is a right R-module, then


 tn®f Io ® g
is an exact sequence ofabelian groups. An analogous state1nent holds for an exact se­
quence in the first variable.

216. Theorem 5.5（定理）

Theorem 5.5. Let R and S be rings and sAn, nB, Cn, RDs (bi)modules as indicated.
 (i) A ®n B is a left S-module surh that s(a ® b) = sa ® b for all s E S, a E A,
b E B.
 (ii) If f : A ----* A' is a hon1omorphism of S-R bilnodules and g : B --7 B' is an
R-1nodu/e luunomorphism, then the induced map f ® g : A ® n B ----* A' ® n B' is a
homotnorphism of left S-modules.
 (iii) C @n D is a right S-n1odule such that (c @ d)s = c @ ds for all c E C,
d E D, s E S .
 (iv) lf h : C � C' is an R-module hon10n1orphism and k : D � D' a homon1or­
phism oj· R-S binzodules, then the induced n1ap h (8) k : C @R D � C' @n D' is a
hnn1omorphisn1 ofright S-modules.

217. Theorem 5.6（定理）

Theorem 5.6. If A,B,C are 1nodules ocer a conunutatice ring R and g : A X B -7 C
is a bilinear map, then there is a unique R-module homomorphism g : A @R B � C
such that gi = g, where i : A X B � A @R B is the canonical bilinear map. The
1nodule A @n B is uniqtte(v determined up to isomorphisn1 by this properry.

218. Theorem 5.7（定理）

Theorem 5.7. lf R is a ring with identity and AR, RB are unitary R-modules, then
there are R-module isomorphisms
 A @R R '"'-� A and R @R B "'"' B.

219. Theorem 5.8（定理）

Theorem 5.8. lfR and S are rings and AR, RBs, 8C are (bi)modules, then there is an
isomorphism
 (A @n B) @s C '"'-� A @R (B @s C).

220. Theorem 5.9（定理）

Theorem 5.9. Let R be a ring, A and { Ai I i E I } right R-modules, B and { Bj I j E J }
left R-1nodules. Then there are group isomorphisms.
 <L Ai) @R B r-v L (Ai @R B) ;
 �I �I

221. Theorem 5.10（定理）

Theore m 5.10. (Adjoint Associativity) Let R and S be rings and An, RBs, Cs (bi)­
modules. Then therl!_ is an isomorphis1n ofabelian groups
 a : Homs(A @n B,C) r-.v HomR(A,Homs(B,C)),
 ,
defined for each f : A @R B ---+ C by
 [(af)(a)](b) = f(a @ b).

 Note that HomR(_,_) and Horns(_,_) consist of homomorphisms of right
modules. Recall that the R-module structure of Homs(B,C) is given by: (gr)(b) =
g(rb) (for r e R, b e B, g e Homs (B,C); see Exercise 4.4(c)).
 '

222. Theorem 5.11（定理）

Theorem 5.11. Let R be a ring with identity. IfA is a unitary right R-module and F
is a free left R-module with basis Y, then every element u of A (8)R F may be written
 n

uniquely'in the fonn u = L ai (8) Yi, where ai e A and the Yi are distinct elementsofY.

 t m

 REMARK. Given u = L ak ®. Yk and v = L bi (8) zi (ak,bi e A , Yk,Z1 e Y),
 k=l j= l
we may, if necessary, insert terms of the form 0 Q9 y (y e Y) and assume that
 n n

u = L ai (8) Yi and v = L bi (8) Yi· The word "uniquely" in Theorem 5.1 1 means
 i=l i=l
 n n n

that if L ai Q9 Yi = L bi (8) Yi, then Ui = bi for every i. In particular, if L ai (8) Yi
 i=l i= l i= l
 n

 0 = L 0 (8) Yi, then a. = 0 for every i.
 i= I
=

223. Corollary 5.12（推论）

Corollary 5.12. lfR is a ring with identity and AR and RB are free R-modules with
bases X and Y respectively, then A @a 8 is a free (right) R-module with basis
W = { x ® y I x e X,y e Y J of cardinality I X I I Y I .

 REMARKS. Since R is an R-R bimodule, so is every direct sum of copies of R.
In particular, every free left R-module is also a free right R-module and vice versa.
However, it is not true in general that a free (left) R-module is a free object in the cat­
egory of R-R bimodules (Exercise 12).

224. Corollary 5.13（推论）

Corollary 5.13. Let S be a ring with identity andR a subring ofS that contains Is. IfF
is a free left R-module with basi� X, then S @R F is a free left S-module with basis
 { 1s @ x I x e X } oj·cardinality lXI.

IV.6 Modules over a Principal Ideal Domain

225. Theorem 6.1（定理）

Theorem 6. 1. Let F be a free module over a principal ideal domain R and G a sub­
module ofF. Then G is a free R-module and rank G < rank F.

226. Corollary 6.2（推论）

Corollary 6.2. Let R be a principal ideal domain. /fA is ajinitely generatedR-module
generated by n elements, then every submodule of A may be generated by m elements
with m < n.

227. Corollary 6.3（推论）

Coro l lary 6.3. A unitary module A over a principal ideal domain is free ifand only if
A is projective.

228. Theorem 6.4（定理）

Theorem 6.4. Let A be a left module over an integral domain R and for each a e A
let Ga = j r e R I ra = 0 J .
 (i) Ga is an ideal of R for each a e A.
 (ii) At = { a e A I Ga � 0} is a submodu/e of A.
 (iii) For each a e A there is an isomorphism of left modules
 Rj Oa '"'"' Ra = { ra I r e R } .
Let R be a principal ideal domain and p e R a prime.
 (iv) /fpia = 0 (equivalently (p i) c 0a), then e. = (pj) with 0 < j < i.
 (v) If t'>a = (p i), then pj a � 0 for all j such that 0 < j < i.

 REMARK. Prime and irreducible elements coincide in a principal ideal domain
by Theorem III. 3. 4.

229. Theorem 6.5（定理）

Theorem 6.5. A finitely generated torsion-free module A over a principal ideal do­
main R is free.
 REMARK. The hypothesis that A is finitely generated is essential (Exercise
Il. l . l 0).

230. Theorem 6.7（定理）

Theore m 6.7. Let A be a torsion module over a principal ideal domain R and for
each prime p e R let A(p) = { a e A I a has order a power of p } .
 (i) A(p) is a submodu/e of A for each prime p e R ;
 (ii) A = L A(p), where the sum is over all primes p e R . If A is finitely gener­
ated, only finitely many of the A(p) are nonzero.

231. Lemma 6.8（引理）

and pn-1 A rf 0 for son1e prime p e R and positice integer n. Let a be an element ofA of
order pn.
 (i) If A � Ra, then there exists a nonzero b e A such that Ra n Rb = 0.
 (ii) There is a submodule C of A such that A = Ra EB C.

 REMARK. The following proof is quite elementary. A more elegant proof of(ii),
which uses the concept of injectivity, is given in Exercise 7.

232. Theorem 6.9（定理）

Theorem 6.9. Let A be a finitely generated module over a principal ideal domain R
such that every element ofA has order a power ofsome prime p e R. Then A is a direct
sum of cyclic R-modules of orders pni , . . . , pnk respectively, where n1 > n2 > · · > ·

nk > 1 .

233. Lemma 6.10（引理）

Lem ma 6. 10. Let A,B, and Ai (i e I) be modules over a principal ideal domain R .
Let r e R and let p e R be prime.

 (i) rA = { ra / a e A I and A[r] = l a e A I ra = 0 } are submodules of A.
 (ii) Rj(p) is a field and A[ p] is a vector space over R/(p) .
 (iii) For each positive integer n there are R-module isomorphisms

 (R/(pn))[p] ""' R j( p) and pm( R/ (pn)) ""' R /(pn-m) (0 < m < n).

 (iv) If A ""' }: Ah then rA ""' }: rA i and A[rJ "'"' }: Ai[rJ.
 ie/ id iEl
 (v) If f : A � B is an R-module isomorphism, then f : At ""' Bt andf : A(p) ""' B(p).

234. Lemma 6.11（引理）

Lemma 6.11. Let R be a principal ideal domain. lf r e R factors as r = Ptn1 • • • Pk nk
with p� , . . . , Pk e R distinct primes and each ni > 0, then there is an R-module iso­
morphism


Consequently every cyclic R-module oforder r is a direct sum ofk cyclic R-modules of"
orders P1 n1, •, Pknk respectively.
 • •

235. Theorem 6.12（定理）

Theorem 6.12. Let A be a finitely generated module over a principal ideal domain R .
 (i) A is the direct sum of a free submodu/e F offinite rank and a finite number of
cyclic torsion modules. The cyclic torsion summands (if any) are of orders r1 , ft ,
 • • • ,



where r 1 , . . , rt are (not necessarily distinct) nonzero nonunit elements of R such that
 .



r1 I r2 l · · I ft. The rank of F and the list ofideals (rt), . . . , (rt) are uniquely determined
 ·




by A.
 (ii) A is the direct sum ofa free submodu/e E offinite rank and a fin ite number of
cyclic torsion modules. Th'3 cyclic torsion summands (ifany) are oforders Pt81, , Pk5k, • • •




where p�, . . . , Pk are (not necessarily distinct) primes in R and s1 , . . . , sk are (not
necessarily distinct) positive integers. The rank ofE and the list of ideals(pt51) , , (p k 8k)
 • • •




are uniquely deterntined by A (except for the order of the Pi)-

 The notation rdr2 l · - l r, means rt divides r2, r2 divides r3, etc. The elements
 ·




r1 , . . . , r, i n Theorem 6. 1 2 are called the invariant factors of the module A just as in
the special case of abelian groups. Similarly Pt81, • •, Pksk are called the elementary
 •




divisors of A .

236. Corollary 6.13（推论）

Corollary 6.13. Two finitely generated modules over a principal ideal domain, A and
B, are isomorphic ifand only ifA/At andB/Bt have the same rank and A and B have
the same invariant factors [resp. elementary divisors].

IV.7 Algebras

237. Definition 7.1（定义）

Defin ition 7 .1. Let K be a commutative ring with identity. A K-algebra (or algebra
over K) A is a ring A such that:

 (i) (A, + ) is a unitary (left) K-module;
 (ii) k(ab) = (ka)b = a(kb) for all k s K and a,b E A.
 A K-algebra A which, as a ring, is a division ring, is called a division algebra.

 The classical theory of algebras deals with algebras over a field K. Such an
algebra is a vector space over K and hence various results of linear algebra are ap­
plicable. An algebra over a field K that is finite dimensional as a vector space over K
is called a finite dimensional algebra over K.

 EXAMPLE. Every ring R is an additive abelian group and hence a Z-module. It
is easy to see that R is actually a Z-algebra.

 EXAMPLES. If K is a commutative ring with identity, then the polynomial ring
K [xr, . . . , xn ] and the power series ring K[[x]J are K-algebras, with the respective
K-module structures given in the usual way.

 EXAMPLE. If V is a vector space over a field F, then the endomorphism ring
Homp(V,V) (Exercise 1 .7) is an F-algebra. The F-module structure of HomF(V,V) is
discussed in the Remark after Theorem 4.8.

 EXAMPLES. Let A be a ring with identity and K a subring of the center of A
such that 1 A E K. Then A is a K-algebra, with the K-module structure being given by
multiplication in A. In particular, every commutative ring K with identity is a
K-algebra.

 EXAMPLE. Both the field of complex numbers C and the division ring of real
quaternions (p. 1 1 7) are division algebras over the field R of real numbers.

 EXAMPLE. Let G be a multiplicative group and K a commutative ring with
identity. Then the group ring K( G) (p. 1 1 7) is actually a K-algebra with K-module
structure given by
 (k , ri c K; gi c G).
K( G) is called the group algebra of G over K.

 EXAMPLE. If K is a commutative ring with identity, then the ring MatnK of all
n X n matrices over K is a K-algebra with the K-module action of K given in the
usual way. More generally, if A is a K-algebra, then so is MatnA .

 REMARK. Since K is commutative, every left K-module (and hence every
K-algebra) A is also a right K module with ka = ak for all a c A, k c K. This fact is
implicitly assumed in Theorems 7.2 and 7 .4 below, where tensor products are used.

 The motivation for the next theorem, which provides another means of defining
K algebras, is the fact that for any ring R the unique map R 0z R R, defined on
 --4

a generator r 0 s by r 0 s � rs, is a homomorphi sm of additive abelian groups.
Since rings are simply Z-algebras , this fact is a special case of

238. Theorem 7.2（定理）

Theorem 7 .2. Let K be a commutative ring with identity and A a unitary left
K-module. Then A jj a K-algebra ifand only ifthere exists a K-modu/e homomorphism
1r : A @K A � A such that the diagram




is commutative. In this case the K-a/gebra A has an identity if and only if there is a
K-module homomorphism I : K � A such that the diagram




is commutative, where !,8 are the isomorphisms ofTheorem 5.1.

239. Definition 7.3（定义）

Definition 7 .3. Let K be a commutative ring with identity and A , B K-algebras.
 (i) A subalgebra of A is a subring of A that is also a K-submodu/e of A.
 (ii) A (left, right, two-sided) algebra ideal of A is a (left, right, two-sided) ideal of
the ring A that is also a K-submodule of A.
 (iii) A homomorphism [resp. isomorphism] of K·algebras f : A � B is a ring ho­
momorphism [isomorphism] that is also a K-module homomorphism [isomorphism].

 REMARKS. If A is a K-algebra, an ideal of the ring A need not be an algebra
ideal of A (Exercise 4). If, however, A has an identity, then for all k e K and a e A
 ka = k(IAa) = (k i A)a and ka = (ka)IA = a(k1 A),
with k l A E A . Consequently, for a left [resp. right] ideal J in the ring A,
 kl = (kl A )J c J [resp. kJ = J(kl A) C J]. I

 J



Therefore, if A has an identity, every (left, right, two-sided) ideal is also a (left, righty
two-sided) algebra ideal.
 The quotient algebra of a K-algebra A by an algebra ideal / is now defined in the
obvious way, as are the direct product and direct sum of a family of K-algebras.
 Tensor products furnish another way to manufacture new algebras. We first
observe that if A and B are K-modules, then there is a K-module isomorphism
a : A @K B � B @K A such that a(a @ b) = b ® a (a E A,b E B) ; see Exercise 2.

240. Theorem 7.4（定理）

Theore m 7 .4. Let A and B be algebras [with identity) over a commutative ring K with
identity. Let 1r be the composition
 lA@a @ l B 1rA ®1rB
 (A @K B) @K (A @K B) (A @K A) @K (B @K B) A @K B,
where 1r 7rB are the product maps of A and B respectively. Then A @K B is a K­
 A,

algebra [with identity) with product map 1r .

未编号术语定义候选

下面是正文中由 called / said to / defined / we say 等关键词抽取的未编号术语定义候选，按出现顺序列出；其中部分内容已包含在上面的编号条目或紧邻说明中。

Chapter II: The Structure of Groups

II.1 Free Abelian Groups

• L4594: An abelian group F that satisfies the conditions of Theorem 1 . 1 is called a free

• L4662: ant of F; IX/ is called the rank of F.

  II.2 Finitely Generated Abelian Groups

• L5107: If G is an abelian group, then the subgroup G, defined in Lemma 2.5 is called the

• L5108: torsion subgroup of G. If G = G, then G is said to be a torsion group. If G, = 0, then
• L5109: G is said to be torsion-free. For a complete classification of all denumerable torsion
• L5296: m1 , . . . , m, as in Theorem 2.6 (ii) are called the invariant factors of G. The uniquely

• L5297: determined prime powers as in Theorem 2.6 (iii) are called the elementary divisors

  II.3 The Krull-Schmidt Theorem

• L5494: is an integer n such that Gi = Grrfor all i > n. G is said to satisfy the descending chain

• L5553: G is called a normal endomorphism if af(b)a-• = f(aba-•) for all a,b e. G.

• L5598: An endomorphism jof a group G is said to be nilpotent if there exists a positive

  II.4 The Action of a Group on a Set

• L5801: When such an action is given, we say that G acts on the set S .

• L5810: G is called a (left) translation. If K is another subgroup of G and S is the set of all left
• L5815: is always denoted hxh-1 and not hx. This action of h E H on G is called conjugation by
• L5823: h and the element hxh-1 is said to be a conjugate of x. If K is any subgroup of G and
• L5838: The equivalence classes of the equivalence relation of Theorem 4.2(i) are called
• L5839: the orbits3 of G on S; the orbit of x E S is denoted x. The subgroup Gx is called vari­
• L5843: { gxg-1 I g e G } of x e G is called the conjugacy class of x. If a subgroup H acts on G
• L5847: the subgroup of H fixing K E S, namely { h E H I hKh-1 = K} , is called the normalizer
• L5895: The equation I Gl = L [ G : Ca (xi)] as in Corollary 4.4 (ii) is called the class
• L5950: The automorphism T0 of Corollary 4. 7(i) is called the inner automorphism in­

• L5951: duced by g. The normal subgroup C(G) = Ker T is called the center of G. An element

  II.5 The Sylow Theorems

• L6116: called a p-group. If H is a subgroup of a group G and H is a p-group, H is said to be

• L6187: A subgroup P of a group G is said to be a Sylow p-subgroup (p prime) if P is a

  II.7 Nilpotent and Solvable Groups

• L6620: { aba - tb-1 J a,b c G J is called the commutator subgroup ofG and denoted G’.

• L6622: The elements aba-1b-1 (a,b c G) are called commutators. The commutators only
• L6645: G<i) is called ith derived subgroup of G . This gives a sequence of subgroups of G,

• L6697: and a lemma. A subgroup H of a group G is said to be characteristic [resp. fully in­

  II.8 Normal and Subnormal Series

• L6933: Gi is normal in G for all i is said to be normal.�

• L6957: tainedfrom S by a finite sequence of one-step refinements. A refinement ofS is said to
• L6969: normal subgroups M of G with M � G (such a subgroup N is called a maximal

Chapter III: Rings

III.1 Rings and Homomorphisms

• L7298: The basic concepts in the theory of rings are defined and numerous examples
• L7312: then R is said to be a commutative ring. IfR contains an element lR such that
• L7314: then R is said to be a ring with identity.
• L7319: The additive identity element of a ring is called the zero element and denoted 0.
• L7363: ment c [resp. b) is called a left [resp. right] inverse ofa. An element a e R that is both
• L7364: left and right invertible is said to be invertible or to be a unit.
• L7373: ment is a unit is called a division ring. A field is a commutative division ring.
• L7462: d ao2 + a12 + a-l + a�?·. K is called the division ring of real quaternions. The
• L7471: cable and exponentiation is defined in R . We have for each a e R and n e N*,
• L7477: Subtraction in a r i n g R is defined in the usual way: a - = + ( - b). Clearly b a
• L7555: phism R —.. R is called an automorphism of R.

• L7584: for all a z R, then R is said to have characteristic n. If no such n exists R is said to

  III.2 Ideals

• L7818: then S is called a subring of R . A subring I of a ring R is a left ideal provided

• L7858: REI\ lARKS. A [left] ideal / of R such that I ¢ 0 and I ¢ R is called a proper [left]
• L7887: [leftJ ideals in R which contain X . Then n Ai is called the [lefr] ideal generated by X.
• L7901: ideal (x) generated by a single element is called a principal ideal. A principal ideal ring
• L8031: The map 1r is called the canonical epimorphism (or projection).
• L8324: II Ri is called the (external) direct product of the family of rings { R1 I i I} . If the €
• L8385: rem 2.24, then R is said to be the (internal) direct product of the ideals A i . As in the

• L8407: Let A be an ideal in a ring R and a,b E R. The element a is said to be congruent to b

  III.3 Factorization in Commutative Rings

• L8710: are said to be associates if a I b and b I a.

• L8931: A Euclidean ·ing which is an integral domain is called a Euclidean domain.

• L8996: greatest common divisor, then a. ,a2 , an are said to be relatively prime.

  III.4 Rings of Quotients and Localization

• L9175: is easily seen to be an equivalence relation. Q is defined to be the set of equivalence

• L9177: denoted ajb and addition and multiplication are defined in the usual way. One
• L9289: The ring s-IR in Theorem 4.3 is called the ring of quotients or ring of fractions or
• L9292: is called the quotient field of the integral domain R. Thus if R = Z, the quotient field
• L9393: REMARKS. S-1 / is called the extension of I in S-1 R. Note that r/s e S-1 / need

• L9483: s-• R is called the localization of R at P and is denoted Rp. If I is an ideal in R, then

  III.5 Rings of Polynomials and Formal Power Series

• L9632: The ring R[x] of Theorem 5 . 1 is called the ring of polynomials over R. Its elements

• L9633: are called polynomials. The notation R[x] is explained below. I n view of Theorem
• L9687: If f = L: aixi E R [x], then the elements ai e R are called the coefficients of f. The
• L9689: element ao is called the �constant term. Elements of R , which all have the form
• L9692: r = (r, 0, 0, . . . ) = rx0 are called constant polynomials. If f = L: aixi = ao +
• L9694: atX + · · · + lln.Xn = anxn + · · · + a1x + a0 has an -¢ 0, then an is called the leading
• L9698: of R[x] is called an indeterminate. One speaks of polynomials in the indeterminate x.
• L9740: The ring R[x�, . . . , xn] of Theorem 5.3 is called the ring of polynomials in n in­
• L9757: i = 1 ,2, . . . , n let xi E R[x 1 , . . . , X n] be defined by xi(Ei) = 1R and xi(u) = 0 for u ‘# Ei .
• L9803: Theorem 5 .4 are called indeterminates. As in the case of one indeterminate symbols
• L9805: The elements ao,a., . . . , am in Theorem 5.4(v) are called the coefficients of the poly-
• L9806: nomial f. A polynomial of the form axt�-.-1xl2 • • x,/‘n (a e. R) is called a monomial in
• L9846: with exponent zero. Then cpf(s�,s2 , . . . , sn) is defined to be L cp(ai)�’1 • - �’” e. S;
• L9947: S of Corollary 5.6 is called the evaluation
• L10042: The ring Rx of Proposition 5.8 is called the ring of formal power series over the
• L10043: ring R. Its elements are called power series. I f R has an identity then the polynomial
• L10044: x = (0, 1 n,O, . . . ) E Rx is called an indeterminate. It is easy to verify that xir = rx i
• L10054: (a0,a�, . .) E Rx is denoted by the formal sum L aixi . The elements ai are called

• L10060: coefficients and a0 is called the constant term. Just as in the case of polynomials this

  III.6 Factorization in Polynomial Rings

• L10293: degree k, is said to be homogeneous of degree k . Recall that for each k ( 1 < k < n),

• L10461: Then (Ct,C2, , C n) is said to be a root or zero of f (or a solution of the polynomial
• L10556: where g(x) e R[xJ and x - c .( g(x) (that is, g(c) � 0). The integer m is called the
• L10557: multiplicity of the root c o f f. If c has multiplicity 1 , c is said to be a simple root. If c
• L10558: has multiplicity nt > 1 , c is called a multiple root. In order to determine when a poly­
• L10581: The polynomial f’ is called the formal derivative of f The word “formal” em­
• L10646: and C( f) is a unit in D, then f is said to be primitive. Clearly for any polynomial

Chapter IV: Modules

IV.1 Modules, Homomorphisms and Exact Sequences

• L11074: then A is said to be a unitary R-module. /fR is a division ring, then a unitary R-module
• L11075: is called a (left) vector space.
• L11077: A (unitary) right R-module is defined similarly via a function A X R -+ A de­
• L11137: If R is a division ring, then an R-module homomorphism is called a linear trans­
• L11140: When the context is clear R-module homomorphisms are called simply homo­
• L11168: b c B. A submodule of a vector space over a division ring is called a subspace.
• L11192: all submodules of A containing X is called the submodule generated by X (or spanned
• L11195: If X is finite, and X generates the module B, B is said to be finitely generated. If
• L11197: X = { a } , then the submodule generated by X is called the cyclic (sub)module gen­
• L11201: generated by X = U Bi is called the sum of the modules Bi. If the index set I is finite,
• L11240: The map 1r is called the canonical epimorphism (or projection).
• L11329: II Ai is called the (external) direct product of the family of R-modules { Ai I i e I J
• L11335: The maps 1rk [resp. c.k] are called the canonical projections [resp. injections].
• L11448: A module A is said to be the (internal) direct sum of a family of submodules
• L11498: morphism, then A/Ker /[resp. B/Im f] is called the coimage of f [resp. cokernel off]
• L11513: quence of the form 0 A � B .!!….:. C � 0 is called a short exact sequence ; note that f

• L11566: Two short exact sequences are said to be isomorphic i f there is a commutative

  IV.2 Free Modules and Vector Spaces

• L11804: A subset X of an R-module A is said to be linearly independent provided that for

• L11811: A set that is not linearly independent is said to be linearly dependent. If A is generated
• L11812: as an R-module by a set Y, then we say that Y spans A. If R has an identity and A is
• L11820: pendent subset of A that spans A is called a basis of A. Observe that the empty set is
• L11840: which satisfies the equivalent conditions of Theorem 2 . 1 , is called a free R-module on
• L12143: two bases of F haoe the same cardinality. Then R is said to have the invariant dimension
• L12144: property and the cardinal number ofany basis of F is called the dimension (or rank) of

• L12252: of dimension. A vector space V over a division ring D is said to be finite dimensional

  IV.3 Projective and Injective Modules

• L12492: especially useful in a categorical setting since they are defined solely in terms of

• L12757: An abelian group D is said to be divisible if given any y E D and 0 � n E Z, there

  IV.4 Hom and Duality

• L13208: is a right R-module by Theorem 4.8(i). HomR(A,R) is called the dual module of A

• L13263: (Theorem 2.1). In part (ii), { fx I x e X } is called the dual basis to X. This theorem is
• L13314: 4(a)). A ** is called the double dual of A.

• L13323: A module A such that () : A � A** is an isomorphism is said to be reflexive.

  IV.5 Tensor Products

• L13496: The quotient group F/K is called the tensor product of A and B ; it is denoted A @R B

• L13705: be bilinear. In this context i is called the canonical bilinear map.

  IV.6 Modules over a Principal Ideal Domain

• L14188: if Gi+I = Gi (that is, Gi+tl G, = 0). Thus b, e: G is defined for each i e: /. Let

• L14260: called the order ideal of a e A . The submodule At in Theorem 6.4 is called the
• L14261: torsion submodule of A . A is said to be a torsion module if A _ = A , and to be torsion­
• L14264: principal ideal of R, say 0a = (r), and a is said to have order r. The element r is
• L14266: Ra generated by a (Theorem 1 .5) is said to be cyclic of order r. Theorem 6.4(iii) shows
• L14613: r1 , . . . , r, i n Theorem 6. 1 2 are called the invariant factors of the module A just as in
• L14614: the special case of abelian groups. Similarly Pt81, • •, Pksk are called the elementary

• L14641: units are ± 1 and primes are defined to be positive. In an arbitrary principal ideal

  IV.7 Algebras

• L14721: A K-algebra A which, as a ring, is a division ring, is called a division algebra.

• L14726: is called a finite dimensional algebra over K.
• L14751: K( G) is called the group algebra of G over K.
• L14796: The homomorphism 1r of Theorem 7.2 is called the product map of the K-algebra
• L14797: A. The homomorphism I is called the unit map.
• L14840: product is defined to be
• L14848: The K-algebra A @K B of Theorem 7.4 is called the tensor product of the K­