5.7 The Krull-Schmidt Theorem

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A useful tool in this section is Exercise 9 of Section 5.2.

Lemma 5.7.1 Let N

μ!

M _! N be R-module homomorphisms with _μ an

automorphism of N. Then M = μN _ ker _.

The following result should remind you of Exercise 3 of Section 4.

Lemma 5.7.2 Let M be an R-module and let f 2 EndR(M).

(i) If M is Artinian and f is injective, then f is surjective.

(ii) If M is Noetherian and f is surjective, then f is injective.

Lemma 5.7.3 (Fitting’s Lemma) Let the R-module M satisfy both chain

conditions, and let f 2 EndR(M). Then for some positive integer n,

M = fnM _ ker fn.

Definition. An R-module M is called indecomposable if it cannot be

written as M = M1 _M2 for nontrivial proper submodules M1,M2 _

M.

Corollary 5.7.3.1 LetM be an indecomposable R-module satisfying both

chain conditions, and let f 2 EndR(M). Then either f is nilpotent or f is

an automorphism.

Corollary 5.7.3.2 LetM be an indecomposable R-module satisfying both

chain conditions. If f1, f2 2 EndR(M), and if g = f1 + f2 is an automorphism,

then one of f1, f2 is an automorphism.

Corollary 5.7.3.3 If M is indecomposable and satisfies both chain conditions,

then EndR(M) is a local ring (i.e., has a unique maximal ideal).

Lemma 5.7.4 Let M = M1 _ M2, N = N1 _ N2 be Artinian R-modules,

and let _ : M ! N be an R-module isomorphism. Write _(m1, 0) =

(_(m1), _(m1)), where _ 2 HomR(M1,N1), _ 2 HomR(M1,N2). If _ :

M1

_=!

N1, then M2

_= N2.

5.7. THE KRULL-SCHMIDT THEOREM 133

Theorem 5.7.5 (Krull-Schmidt Theorem) Let the R-moduleM be both

Noetherian and Artinian, and assume that we are given decompositions

M = M1 _M2 _ · · · _Mr,

M = N1 _ N2 _ · · · _ Ns,

Where each Mi and each Nj is indecomposable. Then r = s, and, possibly

after renumbering, Mi

_= Ni, i = 1, 2 . . . , r.

Exercises

1. Let M be an irreducible R-module. Prove that E = EndR(M) is a

division ring , i.e., each non-zero element of E is invertible. (This

simple result is known as Schur’s Lemma.)

2. Let M be an indecomposable R-module with a composition series 0 _

M1 _ M2 _ · · · _ Mr = M. Assume that the composition factors are

pairwise nonisomorphic. Prove that EndR(M) is a division ring. (Hint:

let _ 2 EndR(M) with _(M) 6= M. Argue that ker _ \ _(M) 6= 0.

Thus ker _ and _(M) share a composition factor. Now what?)

3. Note that the Z-module Z is an indecomposable module which is not

irreducible. Give some other examples.

4. Let V be an F-vector space and let T 2 EndF(V ) be a semisimple

linear transformation (see Exercise 6 of Section 5.5). Prove that the

F[x]-module V is irreducible if and only if it is indecomposable.

5. Let V be an F-vector space and let T 2 EndF(V ). Prove that the F[x]-

module V is indecomposable if and only if V is cyclic and mT (x) =

p(x)e, where p(x) 2 F[x] is irreducible and e is a positive exponent.

6. Let F be a field and let R be the ring

R =

__

a b

0 c

_

| a, b, c 2 F

_

.

R acts in the obvious way on the vector space M, where

M =

__

a

b

_

| a, b 2 F

_

.

Prove thatM is not an irreducible R-module, but it is indecomposable.

134 CHAPTER 5. MODULE THEORY

7. Let R and M be as above and let

L =

__

a

0

_

| a 2 F

_

.

Prove that 0 _ L _ M is a composition series for the R-module

M whose composition quotients are non-isomorphic (i.e., L 6_= M/L).

Conclude from Exercise 2 that EndR(M) is a division ring.

8. Using the Krull-Schmidt Theorem, prove that the elementary divisors

of a finitely generated torsion R-module (where R is a p.i.d.) are

unique. (See Exercise 4 of Section 5.6.)

5.8. INJECTIVE AND PROJECTIVE MODULES 135

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