Page 175 - Advanced Linear Algebra

Modules Over a Principal Ideal Domain 159

We leave proof of the following as an exercise.

Theorem 6.18 Let 4 be a finitely generated torsion module over a principal
ideal domain. The following are equivalent:
)
1 4 is indecomposable
)
2 4 is primary cyclic
)
3 4 has only one elementary divisor:

ElemDiv²4³ ~ ¸ ¹
Thus, the primary cyclic decomposition of 4 is a decomposition of 4 into a
direct sum of indecomposable modules. Conversely, if

4~ ( l Ä l (
is a decomposition of 4 into a direct sum of indecomposable submodules, then
is primary cyclic and so this is the primary cyclic
each submodule (
decomposition of 4 .
Indecomposable Submodules of Prime Order
Readers acquainted with group theory know that any group of prime order is
cyclic. However, as mentioned earlier, the order of a module corresponds to the
smallest exponent of a group, not to the order of a group. Indeed, there are
modules of prime order that are not cyclic. Nevertheless, cyclic modules of
prime order are important.

Indeed, if 4 is a finitely generated torsion module over a principal ideal
domain, with order , then each prime factor of gives rise to a cyclic

submodule > of 4 whose order is and so > is also indecomposable.
Unfortunately, > need not be complemented and so we cannot use it to
decompose 4 . Nevertheless, the theorem is still useful, as we will see in a later
chapter.
Theorem 6.19 Let 4 be a finitely generated torsion module over a principal

ideal domain, with order . If is a prime divisor of , then 4 has a cyclic
(equivalently, indecomposable ) submodule > of prime order .

Proof. If ~ , then there is a # 4 for which $~ #£ but $~ .

Then > ~ ºº$»» is annihilated by and so ²$³ “ . But is prime and

²$³ £ and so ²$³ ~ . Since > has prime order, Theorem 6.18 implies
that > is cyclic if and only if it is indecomposable.
Exercises
1. Show that any free module over an integral domain is torsion-free.
2. Let 4 be a finitely generated torsion module over a principal ideal domain.
Prove that the following are equivalent:
a) 4 is indecomposable
b) 4 has only one elementary divisor (including multiplicity)