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Topics in Module Theory Chapter 7 Topics in Module Theory This chapter will be concerned with collecting a number of results and construc- tions concerning modules over (primarily) noncommutative rings that will be needed to study group representation theory in Chapter 8. 7.1 Simple and Semisimple Rings and Modules In this section we investigate the question of decomposing modules into \simpler" modules. (1.1) De¯nition. If R is a ring (not necessarily commutative) and M 6= h0i is a nonzero R-module, then we say that M is a simple or irreducible R- module if h0i and M are the only submodules of M. (1.2) Proposition. If an R-module M is simple, then it is cyclic. Proof. Let x be a nonzero element of M and let N = hxi be the cyclic submodule generated by x. Since M is simple and N 6= h0i, it follows that M = N. ut (1.3) Proposition. If R is a ring, then a cyclic R-module M = hmi is simple if and only if Ann(m) is a maximal left ideal. Proof. By Proposition 3.2.15, M =» R= Ann(m), so the correspondence the- orem (Theorem 3.2.7) shows that M has no submodules other than M and h0i if and only if R has no submodules (i.e., left ideals) containing Ann(m) other than R and Ann(m). But this is precisely the condition for Ann(m) to be a maximal left ideal. ut (1.4) Examples. (1) An abelian group A is a simple Z-module if and only if A is a cyclic group of prime order. 396 Chapter 7. Topics in Module Theory (2) The hypothesis in Proposition 1.3 that M be cyclic is necessary. The 2 Z-module A = Z2 has annihilator 2Z but the module A is not simple. (3) Consider the vector space F 2 (where F is any ¯eld) as an F [x]-module 2 via the linear transformation T (u1; u2) = (u2; 0). Then F is a cyclic F [X]-module, but it is not a simple F [X]-module. Indeed, F 2 = F [X] ¢ (0; 1) but N = f(u; 0) : u 2 F g is an F [X]-submodule of F 2. Thus the converse of Proposition 1.2 is not true. (4) Let V = R2 and consider the linear transformation T : V ! V de¯ned by T (u; v) = (¡v; u). Then the R[X]-module VT is simple. To see this let w = (u1; v1) 6= 0 2 V and let N be the R[X]-submodule of VT generated by w. Then w 2 N and Xw = T (w) = (¡v1; u1) 2 N. Since any (x; y) 2 V can be written as (x; y) = ®w + ¯Xw where 2 2 2 2 ® = (xu1 + yv1)=(u1 + v1) and ¯ = (yu1 ¡ xv1)=(u1 + v1), it follows that N = VT and hence VT is simple. (5) Now let W = C2 and consider the linear transformation T : W ! W de¯ned by T (u; v) = (¡v; u). Note that T is de¯ned by the same for- mula used in Example 1.4 (4). However, in this case the C[X]-module WT is not simple. Indeed, the C-subspace C ¢ (i; 1) is a T -invariant subspace of W , and hence, it is a C[X]-submodule of WT di®erent from W and from h0i. The following lemma is very easy, but it turns out to be extremely useful: (1.5) Proposition. (Schur's lemma) (1) Let M be a simple R-module. Then the ring EndR(M) is a division ring. (2) If M and N are simple R-modules, then HomR(M; N) 6= h0i if and only if M and N are isomorphic. Proof. (1) Let f 6= 0 2 EndR(M). Then Im(f) is a nonzero submodule of M and Ker(f) is a submodule of M di®erent from M. Since M is simple, it follows that Im(f) = M and Ker(f) = h0i, so f is an R-module isomorphism and hence is invertible as an element of the ring EndR(M). (2) The same argument as in (1) shows that any nonzero homomor- phism f : M ! N is an isomorphism. ut We have a second concept of decomposition of modules into simpler pieces, with simple modules again being the building blocks. (1.6) De¯nition. If R is a ring (not necessarily commutative), then an R- module M is said to be indecomposable if it has no proper nontrivial com- 7.1 Simple and Semisimple Rings and Modules 397 plemented submodule M1, i.e., if M = M1 © M2 implies that M1 = h0i or M1 = M. If M is a simple R-module, then M is also indecomposable, but the converse is false. For example, Z is an indecomposable Z-module, but Z is not a simple Z-module; note that Z contains the proper submodule 2Z. One of the major classes of modules we wish to study is the following: (1.7) De¯nition. An R-module M is said to be semisimple if it is a direct sum of simple R-modules. The idea of semisimple modules is to study modules by decomposing them into a direct sum of simple submodules. In our study of groups there was also another way to construct groups from simpler groups, namely, the extension of one group by another, of which a special case was the semidirect product. Recall from De¯nition 1.6.6 that a group G is an extension of a group N by a group H if there is an exact sequence of groups 1 ¡! N ¡! G ¡! H ¡! 1: If this exact sequence is a split exact sequence, then G is a semidirect product of N and H. In the case of abelian groups, semidirect and direct products coincide, but extension of N by H is still a distinct concept. If G is an abelian group and N is a subgroup, then the exact sequence h0i ¡! N ¡! G ¡! H ¡! h0i is completely determined by the chain of subgroups h0i ⊆ N ⊆ G. By allowing longer chains of subgroups, we can consider a group as obtained by multiple extensions. We will consider this concept within the class of R-modules. (1.8) De¯nition. (1) If R is a ring (not necessarily commutative) and M is an R-module, n then a chain of submodules of M is a sequence fMigi=0 of submodules of M such that ½ ½ ½ ½ (1:1) h0i = M0 6= M1 6= M2 6= ¢ ¢ ¢ 6= Mn = M: The length of the chain is n. m n (2) We say that a chain fNjgj=0 is a re¯nement of the chain fMigi=0 if each Nj is equal to Mi for some i. Re¯nement of chains de¯nes a partial order on the set C of all chains of submodules of M. (3) A maximal element of C (if it exists) is called a composition series of M. 398 Chapter 7. Topics in Module Theory (1.9) Remarks. (1) Note that the chain (1.1) is a composition series if and only if each of the modules Mi=Mi¡1 (1 · i · n) is a simple module. (2) Our primary interest will be in decomposing a module as a direct sum n of simple modules. Note that if M = ©i=1Mi where Mi is a simple R-module, then M has a composition series Mn ½ ½ ½ ½ h0i 6= M1 6= M1 © M2 6= ¢ ¢ ¢ 6= Mi = M: i=1 1 On the other hand, if M = ©i=1Mi, then M does not have a compo- sition series. In a moment (Example 1.10 (2)) we shall see an example of a module that is not semisimple but does have a composition se- ries. Thus, while these two properties|semisimplicity and having a composition series|are related, neither implies the other. However, our main interest in composition series is as a tool in deriving results about semisimple modules. (1.10) Examples. (1) Let D be a division ring and let M be a D-module with a basis fx1; : : : ; xmg. Let M0 = h0i and for 1 · i · n let Mi = hx1; : : : ; xii. n Then fMigi=0 is a chain of submodules of length n, and since Mi=Mi¡1 = hx1; : : : ; xii=hx1; : : : ; xi¡1i » = Dxi =» D; we conclude that this chain is a composition series because D is a simple D-module. (2) If p is a prime, the chain ½ ½ h0i 6= pZp2 6= Zp2 is a composition series for the Z-module Zp2 . Note that Zp2 is not semisimple as a Z-module since it has no proper complemented sub- modules. n (3) The Z-module Z does not have a composition series. Indeed, if fIigi=0 is any chain of submodules of length n, then writing I1 = ha1i, we can properly re¯ne the chain by putting the ideal h2a1i between I1 and I0 = h0i. (4) If R is a PID, then essentially the same argument as Example 1.10 (3) shows that R does not have a composition series as an R-module. 7.1 Simple and Semisimple Rings and Modules 399 (1.11) De¯nition. Let M be an R-module. If M has a composition series let `(M) denote the minimum length of a composition series for M. If M does not have a composition series, let `(M) = 1. `(M) is called the length of the R-module M. If `(M) < 1, we say that M has ¯nite length. Note that isomorphic R-modules have the same length, since if f : M ! N is an R-module isomorphism, the image under f of a composition series for M is a composition series for N.
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