Modules over a ring are a generalization of abelian groups (which are modules over Z). They are basic in the further study of algebra. Section 1 is mostly devoted to carrying over to modules various concepts and results of group theory. Although the classification (up to isomorphism) of modules over an arbitrary ring is quite difficult, we do have substantially complete results for free modules over a ring (Section 2) and finitely generated modules over a principal ideal domain (Section 6). Free modules, of which vector spaces over a division ring are a special case, have widespread applica
tions and are studied thoroughly in Section 2. Projective modules (a generalization of free modules) are considered in Section 3 ; this material is needed only in Section VIII. 6 and Chapter IX.
With the exception of Sections 2 and 6, we shall concentrate on external struc
tures involving modules rather than on the internal structure of modules. Of particu
lar interest are certain categorical aspects of the theory of modules : exact sequences (Section 1 ) and module homomorphisms (Section 4). In addition we shall study vari
ous constructions involving modules such as the tensor product (Section 5). Algebras over a commutative ring
K
with identity are introduced in -section 7.The approximate interdependence of the sections of this chapter is as follows :
A broken arrow A - -� B indicates that an occasional result from Section A is used in Section B, but that Section B is essentially independent of Section A.
168
1. MODU LES, H OM O M O R PH ISMS A N D .EXACT SEQ U E N CES 169
1. MODU LES, HOMOMORPH ISMS AN D EXACT SEQUENCES
Modules over a ring are a generalization of abelian groups (which are modules over Z). Consequently, the first part of this section is primarily concerned with carrying over to modules various concepts and results of group theory. The re
mainder of the section presents the basic facts about exact sequences.
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 is 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 R[[x]] are R-rnodules.
170 CHAPTER IV MODU LES
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(rt + 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.
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) fis 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. In fact, one can define epimorphisms and monomorphisms strictly in categorical terms (objects and morphisms only - no elements) ; see Exercise 2 .
1. MODU LES, HOM O M O R PH ISMS A N D EXACT SEQU ENCES 171
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 (Exercise 3). Similarly if a e A, then Ia = { ra I r E /} is a submodule of A .
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.
iel
Defin ition 1.4. IfX is a subset of a 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 generated by X = U
iel
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.Theorem 1.5. Let R be a ring, A an R-module, X a subset of A, { Bi I i E I } a family ofsubmodules of A and a c A. Let Ra = { ra I r E R } .
(i) Ra is a submodule of A and the n1ap R � Ra 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; Ez}.
li=l j=l
lfR has an identity and A is unitary, then
D = RX =
{
t = lt
riai I s E N*; ai c X ; ri E R}
.172 CHAPTER IV MODU LES
(iv) The sum of the family { Bi I i e I } consists of all finite sums bi1 + · · · + bin with
bik e Bik·
PROOF. Exercise; note that if
R
has an identity lR and A is unitary, then n1 R e R for all n e Z anrl na = (n1 R)a for all a e A. •Theorem 1.6. Let B be a submodule of a 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).
SKETCH OF PROOF OF 1.6. Since A is an additive abelian group, B is a normal subgroup, and A/B is a well-defined abelian group. If a + B = a' + B, then a - a' e B. Since B is a submodule ra - ra' = r(a - a') e B for all r e R. Thus ra + B = ra' + B by Corollary 1.4.3 and the action of R on A/ B is well defined. The remainder of the proof is now easy. •
In view of the preceding results it is not surprising that the various isomorphism theorems for groups (Theorems 1.5.6-1.5 . 1 2) are valid, mutatis mutandis, for modules.
One need only check at each stage of the proof to see that every subgroup or homo
morphism is in fact a submodule or module homomorphism. For convenience we list these results here.
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 if and only iff is an R-module epimorphism and � = Ker f. In particular, A/Ker f ,.-....; lm f.
PROOF. See Theorem 1.5.6 and Corollary 1.5.7. •
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 Kerf C A',
then f is an R -module isomorphism.
PROOF. See Corollary 1.5.8. •
r
1. MODU LES, HOMOMORPH ISMS A N D EXACT SEQU ENCES 173
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) r-...; (B + C)jC ;
(ii) ifC C B, then B/C is a submodule of AjC, and rhere is an R-module isomor
phism (A/C)/(B/C) r-...; A/B.
PROOF. See Corollaries 1.5.9 and 1.5.10. •
Theo rem 1. 10. If R is a ring and B is a submodule of an R-module A, then there is a one-to-one correspondence between the set of all submodules of A containing B and the set of all submodules of A/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.
PROOF. See Theorem 1.5. 1 1 and Corollary 1.5. 1 2 . •
Next we show that products and coproducts always exist in the category of R-modules.
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;, andI:
Ai the direct sum of theiel iel
abelian groups Ai.
(i)
II
Ai is an R-module wirh the action ofR given by r I ai } - { rai I .iei
(ii)
I:
Ai is a submodule ofII
Ai.�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.PROOF. Exercise. •
II
Ai is called the (external) direct product of the family of R-modules { Ai I i e I Jie.I
and
I:
Ai is its (external) direct sum. If the index set is finite, say I = { 1 ,2, . . . , n } ,ie.I
then the direct product and direct sum coincide and will be written A 1 EB A2 EB · · · EB An.
The maps 1rk [resp. c.k] are called the canonical projections [resp. injections].
Tneorem 1.12. /fR is a ring, { Ai I i E I } a family ofR-modules, C an R-module, and I cpi : C � Ai I i E I l a family of R-module homomorphisms, then there is a unique R-module homomorphism cp : C �
II
Ai such that 7rjcp = l{}i for all i E I.II
Ai is�I �I
uniquely determined up to isomorphism by this property. In other words,
II
Ai is aie.I
product in the category o fR -modules.
PROOF. By Theorem 1.8.2 there is a unique group homomorphism cp : C �
I1
Aiwhich has the desired property, given by cp(c) = { cpi(c) }i11• Since each cpi is an R-