5.1 The Basic Homomorphism Theorems

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In Section 4.1 we introduced some of the basics of module theory, as they

were indespensible to our study of Dedekind domains. In the present chapter,

we embark on a more systematic study of module theory; one very important

difference here is that unless otherwise stated, the rings in question

need not be commutative.

There are two basic homomorphism theorems worth mentioning here.

The proofs are entirely routine and mimick the corresponding proofs for

abelian groups (i.e.,Z-modules).

Theorem 5.1.1 (The Fundamental Homomorphism Theorem) Let

R be a ring and let _ : M1 ! M2 be a homomorphism of R-modules. Then

_ admits a factorization, according to the commutative diagram below:

M1 M2

M1/ker _

-

@

@

@R 􀀀

􀀀

􀀀_

_

_ _¯

where _ : M1 ! M1/ker _ is the canonical projection, and where ¯_(m1 +

ker _) = _(m1), m1 2 M1.

105

106 CHAPTER 5. MODULE THEORY

The next result is sometimes also called the Second Isomorphism Theorem.

Theorem 5.1.2 (The Noether Isomorphism Theorem) Let R be a ring,

and let M be an R-module. If M1, M2 are submodules of M, then

(M1 +M2)/M1

_= M2/(M1 \M2).

At the risk of being repetitive, we’ll state the modular law again, as it

is a key ingredient in the “Third Isomorphism Theorem,” below.

Lemma 5.1.3 (Modular Law) Let R be a ring, and let M be an Rmodule.

Assume that M1,M2 and N are submodules of M with M1 _ M2.

Then

M2 + (N \M1) = (M2 + N) \M1.

The next result is considerably more esoteric and is variably called the

Butterfly Lemma, Third Isomorphism Theorem or the Zassenhaus Lemma.

This will be used in proving the Schreier Refinement Theorem; see Proposition

5.6.4, below.

Theorem 5.1.4 Let R be a ring, and let M be an R-module. Assume that

we have submodules N2 _ N1 _ M, M2 _ M1 _ M. Then

M2 + (N1 \M1)

M2 + (N2 \M1)

_=

N2 + (M1 \ N1)

N2 + (M2 \ N1) .

Exercises

1. Let K _ M _ N be R-modules. Prove that (N/K)/(M/K) _= N/M.

2. Give examples of R-modulesM1, M2 such thatM1

_=Z M2, butM1 6_=R

M2.

3. Let M be an R-module and let M1 _ M be a submodule. If _ : M !

N is a homomorphism of R-modules such that ker _ _ M1, prove

that M/M1

_= _M/_M1. Give a counterexample to show that this

hypothesis is necessary.

4. Let R be a Dedekind domain with fraction field E, and let I, J _ E

be fractional ideals representing classes [I], [J] 2 CR, the ideal class

group of R (See Section 4.5). If [I] = [J], prove that I _=R J. (The

converse is also true; see Exercise 12 of Section 7.2.)

5.2. DIRECT PRODUCTS AND SUMS OF MODULES 107

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