5.6 Chain Conditions and Series of Modules

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Recall that in Section 4.1 (see Page 89) we defined a Noetherian module

to be a module M such that if

M1 _ M2 _ · · ·

is a chain of submodules, then there exists an integer N such that if n _ N,

then Mn = MN. In other words, a Noetherian modules is one that satisfies

the ascending chain condition (a.c.c.) on submodules. In a completely analogous

way, we define an Artinian module to be one satisfying the descending

chain condition (d.c.c) on submodules.

For convenience, we remind the reader of the following equivalent conditions

for a module to be Noetherian (See Proposition 4.1.2 of Section 4.1.)

Proposition 5.6.1 The following conditions are equivalent for the R-module

M.

(i) M is Noetherian.

(ii) Every submodule of M is finitely generated.

(iii) Every nonempty collection of submodules of M contains a maximal

element (relative to containment).

As one would expect, Artinian modules can be characterized as follows:

Proposition 5.6.2 The following conditions are equivalent for the R-module

M.

(i) M is Artinian.

(ii) Every nonempty collection of submodules of M contains a minimal

element (relative to containment).

The following is proved very easily, using the Modular Law. (See Lemma 5.1.3,

Page 106.)

Proposition 5.6.3 Let 0 ! K ! M ! N ! 0 be a short exact sequence

of R-modules.

(a) M is Noetherian if and only if both K and N are.

130 CHAPTER 5. MODULE THEORY

(b) M is Artinian if and only if both K and N are.

Let M 6= 0 be an R-module. We say that M is irreducible (or is simple)

if M contains no nontrivial submodules. Here are a few examples:

1. An irreducible Z-module is simply a cyclic group of prime order.

2. The ring Z contains no nontrivial ideals that are also irreducible as

Z-modules.

3. Let R be the ring M2(F) of 2-by-2 matrices over a field F. Let M be

the “natural” R-module

M =

__

a

b

_

| a, b 2 F

_

.

Then M is an irreducible R-module.

4. Let V be an F-vector space, and let T 2 EndF(V ). If V is an F[x]-module

in the usual way, then V is irreducible if and only if the minimal

polynomial mT (x) is irreducible, and deg mT (x) = dim V.

Let M be an R-module. A chain of submodules of M

0 = M0 _ M1 _ M2 _ · · · _ Mr = M

is called a composition series if for each i _ 1, Mi/Mi−1 is an irreducible

R-module.

Proposition 5.6.4 (Schreier Refinement Theorem) Let N _ M be

R- modules, and consider two chains of submodules:

N = M0 _ M1 _ · · · _ Mr = M,

N = N0 _ N1 _ · · · _ Ns = M.

Then both chains can be refined so that the resulting chains have the same

length and isomorphic factors (in some order).

Corollary 5.6.4.1 (Jordan-H¨older Theorem) LetM be an R-module

with two composition series

0 = M0 _ M1 _ · · · _ Mr = M,

0 = N0 _ N1 _ · · · _ Ns = M.

Then r=s and in some order, the successive factors are isomorphic.

5.6. CHAIN CONDITIONS AND SERIES OF MODULES 131

Of course, the above theorem can also be proved as in the proof of

Theorem 1.7.4 of Section 1.7. However, the Schreier Refinement Theorem

gives a different approach.

Theorem 5.6.5 The R-module M has a composition series if and only if it

is both Noetherian and Artinian.

Exercises

1. State and prove the appropriate version of the Butterfly Lemma for

groups.

2. Prove that the Z-module Q is neither Noetherian nor Artinian.

3. Show that the Z-module Z(p1) p prime is Artinian but not Noetherian.

(See Exercise 5 of Section 5.3.)

4. Let R be a principal ideal domain and let M be a finitely generated

torsion R-module. Prove that M is both Artinian and Noetherian.

What if M is torsion-free?

132 CHAPTER 5. MODULE THEORY

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