Page 169 - Advanced Linear Algebra

Modules Over a Principal Ideal Domain 153

Then
4 ~ ºº# »» l Ä l ºº# »»

and
4 ~ ºº" »» l Ä l ºº" »»

!
But ºº# »» ~ ºº # »» is a cyclic submodule of 4 with annihilator º c » and so

by the induction hypothesis
~ ! and ~ ÁÃÁ ~

which concludes the proof of uniqueness.
For part 3), suppose that ¢4 š 5 and 4 has annihilator chain
²
²º
ann º# »»³ ‹ Ä ‹ ann ºº# »»³

and has annihilator chain
5
²
²º
ann º" »»³ ‹ Ä ‹ ann ºº" »»³

Then

5 ~ 4 ~ ºº # »» l Ä l ºº # »»

and so ~ and after a suitable reindexing,
ann²ºº# »»³ ~ ann²ºº # »»³ ~ ann²ºº" »»³

Conversely, suppose that
4 ~ ºº# »» l Ä l ºº# »»

and
5 ~ ºº" »» l Ä l ºº" »»

have the same annihilator chains, that is, ~ and
ann ² ann ºº# »»³
²ºº" »»³ ~

Then
9 9
ºº" »» š ~ š ºº# »»

ann ²ºº" »»³ ann ²ºº# »»³

The Primary Cyclic Decomposition
Now we can combine the various decompositions.
(
Theorem 6.13 The primary cyclic decomposition theorem) Let 4 be a
finitely generated torsion module over a principal ideal domain .
9