Page 163 - Advanced Linear Algebra

Modules Over a Principal Ideal Domain 147

(
Definition Let be a prime in . A -primary or just primary) module is a

9
module whose order is a power of .

(
Theorem 6.10 The primary decomposition theorem) Let 4 be a torsion
module over a principal ideal domain , with order
9

~ Ä

where the 's are distinct nonassociate primes in .

9
)
1 4 is the direct sum

4~ 4 l Ä l 4
where

4 ~ 4 ~ ¸#  4 “ # ~ ¹

is a primary submodule of order . This decomposition of 4 into primary

submodules is called the primary decomposition of 4 .
)
2 The primary decomposition of 4 is unique up to order of the summands.
That is, if

4~ 5 l Ä l 5
where 5 is primary of order and Á Ã Á are distinct nonassociate

primes, then ~ and, after a possible reindexing, 5 ~4 . Hence,
~ — and , for ~ Á à Á .
)
3 Two -modules 4 and 5 are isomorphic if and only if the summands in
9
their primary decompositions are pairwise isomorphic, that is, if
4~ 4 l Ä l 4
and

5~ 5 l Ä l 5
are primary decompositions, then ~ and, after a possible reindexing,
for ~ ÁÃÁ .

4 š 5

Proof. Let us write ~° and show first that

4~ 4 ~ ¸ # “ #  4¹

Since ² 4³ ~ 4 ~ ¸ ¹ , we have 4 ‹ 4 . On the other hand, since

and are relatively prime, there exist Á  9 for which

b ~

then
and so if %4