MODULES

Excerpted from Abstract Algebra II, copyright 1996 by John Beachy.
10.4 Composition series
10.5 Semisimple modules
10.6 Tensor products
10.7 Modules over a principal ideal domain

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Composition series

10.4.1. Definition. A composition series of length n for a nonzero module M is a chain of n+1 submodules

M = M0 M1 . . . Mn = (0)

such that Mi-1/Mi is a simple module for i=1,2,...,n. These simple modules are called the composition factors of the series.

10.4.2. Theorem. [Jordan-Holder] If a module M has a composition series, then any other composition series for M is equivalent to it.

As an immediate consequence of the Jordan-Holder theorem, if a module RM has a composition series, then all composition series for M must have the same length, which we denote by (M). This is called the length of the module, and we simply say that the module has finite length. Since any ascending chain of submodules can be refined to a composition series, (M) gives a uniform bound on the number of terms in any properly ascending chain of submodules. We also note that if M1 and M2 have finite length, then
(M1 M2) = (M1) + (M2).

10.4.3. Proposition. A module has finite length if and only if it is both Artinian and Noetherian.

A module RM is said to be indecomposable if its only direct summands are (0) and M. As our first example, we note that Z is indecomposable as a module over itself, since the intersection of any two nonzero ideals is again nonzero. To give additional examples of indecomposable Z-modules, recall any finite abelian group is isomorphic to a direct sum of cyclic groups of prime power order. Using this result, we see that a finite Z-module is indecomposable if and only if it is isomorphic to Zn, where n=pk for some prime p.

10.4.4. Proposition. If RM has finite length, then there exist finitely many indecomposable submodules M1, M2, . . . , Mn such that

M = M1 M2 . . . Mn.

10.4.5. Lemma. [Fitting] Let M be a module with length n, and let f be an endomorphism of M. Then

M = Im (fn) ker(fn) .

10.4.6. Proposition. Let M be an indecomposable module of finite length. Then for any endomorphism f of M the following conditions are equivalent.
(1) f is one-to-one;
(2) f is onto;
(3) f is an automorphism;
(4) f is not nilpotent.

10.4.7. Proposition. Let M be an indecomposable module of finite length, and let f1, f2 be endomorphisms of M. If f1 + f2 is an automorphism, then either f1 or f2 is an automorphism.

10.4.8. Lemma. Let X1, X2, Y1, Y2 be left R-modules, and let f: X1 X2 -> Y1 Y2 be an isomorphism. Let i1 : X1 -> X1 X2 and i2 : X2 -> X1 X2 be the natural inclusion maps, and let p1 : Y1 Y2 -> Y1 and p2 : Y1 Y2 -> Y2 be the natural projections. If p1 f i1 : X1 -> Y1 is an isomorphism, then p2 f i2 : X2 -> Y2 is an isomorphism.

10.4.9. Theorem. [Krull-Schmidt] Let {Xj} and {Yi} be indecomposable left R-modules of finite length. If

X1 . . . Xm Y1 . . . Yn,

then m = n and there exists a permutation Sn with (j)=i and XjYi, for 1jm.

Semisimple modules

10.5.1. Definition. Let M be a left R-module. The sum of all minimal submodules of M is called the socle of M, and is denoted by Soc(M). The module M is called semisimple if it can be expressed as a sum of minimal submodules.

A semisimple module R M behaves like a vector space in that any submodule splits off, or equivalently, that any submodule N has a complement N' such that N+N'=M and NN'=0.

10.5.2. Theorem. Any submodule of a semisimple module has a complement that is a direct sum of minimal submodules.

10.5.3. Corollary. The following conditions are equivalent for a module R M.
(1) M is semisimple;
(2) Soc (M) = M.
(3) M is completely reducible;
(4) M is isomorphic to a direct sum of simple modules.

10.5.4. Corollary. Every vector space over a division ring has a basis.

10.5.5. Definition. The module RQ is said to be injective if for each one-to-one R-homomorphism i:RM->RN and each R-homomorphism f:M->Q there exists an R-homomorphism f*:N->Q such that f*i=f.

10.5.6. Theorem. The following conditions are equivalent for the ring R.
(1) R is a direct sum of finitely many minimal left ideals;
(2) R R is a semisimple module;
(3) every left R-module is semisimple;
(4) every left R-module is projective;
(5) every left R-module is injective;
(6) every left R-module is completely reducible.

10.5.7. Corollary. Let D be a division ring, and let R be the ring Mn(D) of all n×n matrices over D. Then every left R-module is completely reducible.

Let R be a ring, and let G be a group. The group ring RG is defined to be a free left R-module with the elements of G as a basis. The multiplication on RG is defined by
( wG aw w ) ( xG bx x ) = zG cz z where cz = z=wx aw bx.

The crucial property of a group ring is that it converts group homomorphisms from G into the group of units of a ring into ring homomorphisms. To be more precise, let S be a ring,
let :G->Sx be a group homomorphism,
and let :R->Z(S) be any ring homomorphism.
(Recall that Sx denotes the group of invertible elements of S and Z(S) denotes the center of S.)
Then there is a unique ring homomorphism :RG->S such that
(g)=(g) for all gG and (r)=(r) for all rR.

10.5.8. Theorem. [Maschke] Let G be a finite group and let K be a field such that |G| is not divisible by chr(K). Then every KG-module is completely reducible.

10.5.9. Theorem. [Baer's criterion] For any left R-module Q, the following conditions are equivalent.
(1) The module Q is injective;
(2) for each left ideal A of R and each R-homomorphism f:A->Q there exists an extension f*:R->Q such that f*(a)=f(a) for all aA;
(3) for each left ideal A of R and each R-homomorphism f:A->Q there exists qQ such that f(a)=aq, for all aA.

10.5.10. Proposition. Let D be a principal ideal domain, with quotient field Q.
(a) The module DQ is injective.
(b) Let I be any nonzero ideal of D, and let R be the ring D/I. Then R is an injective module, when regarded as an R-module.

Tensor products

We need to begin with the definition of a bilinear function. Given modules MR and RN over a ring R and an abelian group A, a function : M × N -> A is said to be R-bilinear if
(i) ( x1 + x2 , y ) = ( x1 , y ) + ( x2 , y );
(ii) ( x , y1 + y2 ) = ( x , y1 ) + ( x , y2 );
(iii) ( xr, y ) = ( x , ry )
for all x, x1, x2 M, y, y1, y2 N and rR.

10.6.1. Definition. A tensor product of the modules MR and RN is an abelian group T(M,N) and an R-bilinear map : M × N -> T(M,N) such that for any abelian group A and any R-bilinear map : M × N -> A there exists a unique Z-homomorphism f:T(M,N)->A such that f=.
The group T(M,N) is usually denoted by MRN, and for xM, yN the image (x,y) is denoted by xy.

10.6.2. Proposition. Let MR and RN be modules. The tensor product MRN is unique up to isomorphism, if it exists.

10.6.3. Proposition. For any modules MR and RN, the tensor product MRN exists.

10.6.4. Proposition. Let M, M' be right R-modules, let N, N' be left R-modules, and let fHom(MR,M'R) and gHom(RN,RN').
(a) Then there is a unique Z-homomorphism fg:MRN->M'RN'
with (fg) (xy) = f(x) g(y) for all xM, y N.
(b) If f and g are onto, then fg is onto, and ker(fg) is generated by all elements of the form xy such that either xker(f) or yker(g).

10.6.5. Proposition. Let MR be a right R-module, and let {N} I be a collection of left R-modules. Then

M R ( I N) I ( M R N).

10.6.6. Definition. Let R and S be rings, and let U be a left S-module. If U is also a right R-module such that (sx)r = s(xr) for all s S, r R, and x U, then U is called an S-R-bimodule.
If U is an S-R-bimodule we use the notation SUR.

10.6.7. Proposition. Let R, S, and T be rings, and let SUR, RMT , and SNT be bimodules.
(a) The tensor product UR M is a bimodule, over S on the left and T on the right.
(b) The set HomS(U,N) is a bimodule, over R on the left and T on the right.

10.6.8. Proposition. Let M be a left R-module, and let N be a left S-module.
(a) RRM M, as left R-modules.
(b) HomS(S,N) N, as left S-modules.

10.6.9. Proposition. Let SUR be a bimodule. For any modules RM and SN, there is an isomorphism
: HomS( U R M , N) -> HomR( M, HomS (U,N) ).

10.6.10. Corollary. Let R and S be rings, and let :R->S be a ring homomorphism. Let M be any left R-module. Then SRM is a left S-module, and for any left S-module N we have
HomS( S R M, N) HomR(M, N).

Modules over a principal ideal domain

The fundamental theorem of finite abelian groups states that any finite abelian group is isomorphic to a direct product of cyclic groups of prime power order. From our current point of view, a finite abelian group is a module over the ring of integers Z, and in this section we will show that we can extend the fundamental theorem to modules over any principal ideal domain. This includes the ring Q[x] of all polynomials with coefficients in the field Q, and in this case all of the cyclic modules are infinite, so we cannot restrict ourselves to finite modules. The appropriate generalization is to consider finitely generated torsion modules, which we now define. We will also consider finitely generated torsionfree modules, which turn out to be free. In this section all rings will be commutative, and so we simply refer to modules rather than left or right modules.

10.7.1. Definition. Let D be an integral domain, and let M be a D-module.
An element m M is called a torsion element if Ann(m)(0). The set of all torsion elements of M is denoted by tor(M).
If tor(M)=(0), then M is said to be a torsionfree module.
If tor(M) = M, then M is said to be a torsion module.

10.7.2. Proposition. Let D be an integral domain, and let M be a D-module. Then tor(M) is a submodule of M, and M/tor(M) is a torsionfree module.

10.7.3. Lemma. Let D be a principal ideal domain with quotient field Q. Then any nonzero finitely generated submodule of Q is free of rank 1.

10.7.4. Lemma. Let D be a principal ideal domain, and let M be a finitely generated torsionfree D-module. If M contains a submodule N such that N is free of rank 1 and M/N is a torsion module, then M is free of rank 1.

10.7.5. Theorem. If D is a principal ideal domain, then any nonzero finitely generated torsionfree D-module is free.

10.7.6. Proposition. Let D be a principal ideal domain, and let M be a finitely generated D-module. Then either M is torsion, or tor(M) has a complement that is torsionfree. In the second case, M=tor(M)N for a submodule NM such that N is free of finite rank.

The previous proposition shows that to complete the description of all finitely generated modules over a principal ideal domain we only need to characterize the finitely generated torsion modules. The first step is to show that any finitely generated torsion module can be written as a direct sum of finitely many indecomposable modules, and this is a consequence of the next propositions.

10.7.7. Proposition. Let D be a principal ideal domain, and let a be a nonzero element of D. If
a = p1m1 p2m2 . . . pkmk is the decomposition of a into a product of irreducible elements, then we have the following ring isomorphism.

D/aD (D/p1m1D) (D/p2m2D) . . . (D/pkmkD)

10.7.8. Proposition. Let D be a principal ideal domain, let p be an irreducible element of D, and let M be any indecomposable D-module with Ann(M)=pkD. Then M is a cyclic module isomorphic to D/pkD.

10.7.9. Theorem. Let D be a principal ideal domain, and let M be a finitely generated D-module. Then M is isomorphic to a finite direct sum of cyclic submodules each isomorphic to either D or D/pkD, for some irreducible element p of D. Moreover, the decomposition is unique up to the order of the factors.

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