5.3 Modules over a Principal Ideal Domain

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All modules in this section are modules over a principal ideal domain. The

first result shows that rank behaves nicely with respect to submodules.

Proposition 5.3.1 Let M be a free module over the principal ideal domain

R. If N is a submodule of M, then N is free, and rank (N) _ rank (M).

The above result actually characterizes principal ideal domains, as follows

from Exercise 1, below.

Let M be an R-module, R a p.i.d., and let m 2 M. Set Ann (m) = {r 2

R| rm = 0}; note that Ann (m) is an ideal of R. Since R is a p.i.d., we

conclude that Ann (m) = Ra, for some a 2 R. The element a, well defined

up to associates, is called the order of m, and denoted o(m). If o(m) 6= 0,

m is called a torsion element of M. Note that the torsion elements of M

form a submodule of M, called the torsion submodule of M, and is denoted

T(M). If T(M) = 0, M is called a torsion-free R-module. On the other

hand, if every element of M is torsion, then M is called a torsion module.

Finally, note that M/T (M) is a torsion-free module.

Proposition 5.3.2 Let M be a finitely generated torsion-free R-module,

where R is a principal ideal domain. Then M is free.

Note that the condition of finite generation in the above proposition is

crutial since the abelian group (Z-module) Q is torsion-free, but not free.

Proposition 5.3.3 Let M be a finitely generated R-module, where R is a

principal ideal domain. Then M = F _T(M), where F is a free submodule

of M.

From Proposition 5.3.3, it follows that in order to classify finitely generated

modules over a p.i.d., it suffices to classify finitely generated torsion

modules over a p.i.d. Indeed, note that if M is finitely generated over the

p.i.d. R, then by Corollary 4.1.3.1 of Chapter 4, T(M) is also finitely generated.

Let M be an R-module and let r 2 R. Define M[r] = {m 2 M| rm = 0}.

Clearly M[r] is a submodule of M, and that every element of M[r] has order

dividing r. Assume that M is a finitely generated torsion R-module. Then

0 6= Ann (M) := {r 2 R| rM = 0}; since Ann (M) is clearly an ideal of R,

we conclude that Ann (M) = Ra, for some 0 6= a 2 R. The element a 2 R,

116 CHAPTER 5. MODULE THEORY

well defined up to associates, is called the exponent of M, and is sometimes

denoted exp (M). It should be clear that if N _ M then exp (N) divides

exp (M).

Theorem 5.3.4 (Primary Decomposition Theorem) LetM be a finitely

generated torsion module over the principal ideal domain R. Let a be the

exponent of M, and assume that a = pe1

1 pe2

2 · · · pek

k is the factorization of a

into its prime powers. Then

M =

Mk

i=1

M[pei

i ].

Thus the problem of determining the structure of a finitely generated

torsion R-module is reduced to that of determining the structure of a finitely

generated R-module of prime-power exponent.

Recall that an R-module M is called cyclic if it is of the form Rx, for

some x 2 M. It should be clear that if M = Rx, and if o(x) = a, then

exp (M) = a.

Theorem 5.3.5 Let M be a finitely generated R-module over the principal

ideal domain R, and assume that exp (M) = pe, where p is a prime in R.

Then there exists a unique sequence e1 = e _ e2 _ . . . _ el, and cyclic

submodules Z1,Z2, . . . ,Zl, such that M =

Ml

i=1

Zi.

Corollary 5.3.5.1 (Elementary Divisor Theorem) LetM be a finitely

generated torsion R-module over the principal ideal domain R with a =

exp (M) = pe1

1 · · · pek

k (prime factorization). Then there exists unique sequences

ei1 = ei _ ei2 . . . _ eili , i = 1, 2, . . . k

such that

M =

Mk

i=1

Mli

j=1

Zij ,

where each Zij is a cyclic submodule of exponent peij

i .

The prime powers peij

i occurring in the above are often called the elementary

divisors of the torsion module M, and the cyclic submodules Zij

5.3. MODULES OVER A PRINCIPAL IDEAL DOMAIN 117

are called elementary components . Thus, as a simple example, the abelian

group Z25 _ Z5 _ Z3 _ Z3 has (25, 5, 3, 3) as its sequence of elementary

divisors. The cyclic groups Z25,Z5,Z3 occur as elementary components.

For many applications, the decomposition of a finitely generated torsion

module into elementary components is too fine. Indeed, note the following:

Lemma 5.3.6 Let Z1, Z2 be cyclic R-modules of exponents a1, a2, and

assume that a1, a2 are relatively prime in R. Then Z1 _ Z2 is cyclic of

exponent a1a2.

As a result, we have the following.

Theorem 5.3.7 (Invariant Factor Theorem) Let M be a finitely generated

torsion module over the principal ideal domain R, and assume that

exp (M) = a. Then there exists a unique sequence

a1 = a, a2, . . . , ar,

with ai+1|ai, i = 1, . . . , r−1, and cyclic submodules M1, . . .Mr, exp (Mi) =

ai, i = 1, . . . , r, such that M =

Mr

i=i

Mi.

The elements a1, a2, . . . , ar in the above theorem are called the invariant

factors of M.

Exercises

1. Let R be a commutative ring, and assume that every ideal of R is a

free submodule of R. Prove that R is a p.i.d.

2. Let M be a finitely generated free module over the p.i.d. R, and let

N be a submodule of M. Prove that M and N have the same rank if

and only if the quotient module M/N is a torsion module.

3. Classify all the finite abelian groups of order 300.

4. For each prime p, define the subgroup Tp of the additive group of the

rationals by setting

Tp = {a/pi 2 Q| a, i 2 Z}.

Prove that the abelian groups Q, Tp, p is prime are not finitely generated

abelian groups.

118 CHAPTER 5. MODULE THEORY

5. Prove that the torsion abelian groups Q/Z and Z(p1) := Tp/Z, p is prime

are isomorphic to subgroups of the group T, the multiplicative group

of modulus 1 complex numbers.

6. Prove that the torsion abelian group Q/Z admits a primary decomposition

of the form Q/Z =

M

p prime

Z(p1).

7. Prove that every finite subgroup of the abelian group Z(p1) is cyclic.

8. Prove that Z(p1) has no maximal subgroups.

9. Let R be a p.i.d., and let M be a cyclic R-module of exponent a 2 R.

Prove that M is a free R/Ra-module.

10. Let M be a finitely generated torsion module over the p.i.d. R, and let

a 2 R be the exponent of M. Prove that AutR(M) acts transitively

on the elements of order a in M.

5.4. CALCULATION OF INVARIANT FACTORS 119

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