Page 174 - Advanced Linear Algebra

158 Advanced Linear Algebra

annihilators in the primary cyclic decomposition. Note that is an order of 4 ,

that is,
ann²4³ ~ º »

Note also that the product

~ Ä

of the invariant factors of 4 has some nice properties. For example, is the
product of all the elementary divisors of 4 . We will see in a later chapter that
in the context of a linear operator on a vector space, is the characteristic

polynomial of .

Characterizing Cyclic Modules
The primary cyclic decomposition can be used to characterize cyclic modules
via their elementary divisors.

Theorem 6.17 Let 4 be a finitely generated torsion module over a principal
ideal domain, with order
~ Ä

The following are equivalent:
)
1 4 is cyclic.
)
2 4 is the direct sum
4 ~ ºº# »» l Ä l ºº# »»

of primary cyclic submodules ºº# »» of order .

)
3 The elementary divisors of 4 are precisely the prime power factors of :

ElemDiv²4³ ~ ¸ ÁÃÁ ¹

Proof. Suppose that 4 is cyclic. Then the primary decomposition of 4 is a
primary cyclic decomposition, since any submodule of a cyclic module is cyclic.
Hence, 1) implies 2). Conversely, if 2) holds, then since the orders are relatively
prime, Theorem 6.4 implies that 4 is cyclic. We leave the rest of the proof to
the reader.
Indecomposable Modules
The primary cyclic decomposition of 4 is a decomposition of 4 into a direct
sum of submodules that cannot be further decomposed. In fact, this
characterizes the primary cyclic decomposition of 4 . Before justifying these
statements, we make the following definition.

Definition A module 4 is indecomposable if it cannot be written as a direct
sum of proper submodules.