Page 166 - Advanced Linear Algebra

150 Advanced Linear Algebra

or equivalently,

~ ‚ ‚ Ä ‚
)
2 As to uniqueness, suppose that 4 is also the direct sum
4 ~ ºº" »» l Ä l ºº" »»

of cyclic submodules with annihilators ann²ºº" »»³ ~ º » , arranged in

ascending order
ann ² ann ºº" »»³
²ºº" »»³ ‹ Ä ‹

or equivalently

‚ ‚ Ä ‚

Then the two chains of annihilators are identical, that is, ~ and
ann ² ann ºº# »»³
²ºº" »»³ ~

for all . Thus, — and ~ for all .

3 Two -primary -modules
)

9
4 ~ ºº# »» l Ä l ºº# »»

and
5 ~ ºº" »» l Ä l ºº" »»

are isomorphic if and only if they have the same annihilator chains, that is,
if and only if ~ and, after a possible reindexing,
ann ² ann ºº# »»³
²ºº" »»³ ~

Proof. Let # 4 have order equal to the order of 4 , that is,

ann²# ³ ~ ann²4³ ~ º »

Such an element must exist since ²# ³  for all #  4 and if this inequality

is strict, then c will annihilate 4 .
If we show that ºº# »» is complemented, that is, 4 ~ ºº# »» l : for some

is also a finitely generated primary torsion module
submodule : , then since :
over , we can repeat the process to get
9
4 ~ # ºº »» l # ºº »» l :
where ann²# ³ ~ º » . We can continue this decomposition:

4 ~ # ºº »» l # ºº »» l Ä l # ºº »» l :
as long as : £ ¸ ¹ . But the ascending sequence of submodules