Page 165 - Advanced Linear Algebra
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Modules Over a Principal Ideal Domain 149
. Moreover, since
~
4 ¯ ~ ¯ ² ³~ ¯ ~ ¯ 5
.
it follows that ¢4 5
The Cyclic Decomposition of a Primary Module
The next step in the decomposition process is to show that a primary module
can be decomposed into a direct sum of cyclic submodules. While this
decomposition is not unique see the exercises , the set of annihilators is unique,
(
)
as we will see. To establish this uniqueness, we use the following result.
Lemma 6.11 Let 4 be a module over a principal ideal domain and let
9
9 be a prime.
)
1 If 4 ~ ¸ ¹ , then 4 is a vector space over the field 9°º » with scalar
multiplication defined by
² b º »³# ~ #
for all #4 .
)
2 For any submodule of 4 the set
:
: ² ³ ~ ¸# : # ~ ¹
is also a submodule of 4 and if 4 ~ : l ; , then
4 ² ³ ~ : ² ³ l ; ² ³
)
Proof. For part 1 , since is prime, the ideal º » is maximal and so ° 9 º » is a
field. We leave the proof that 4 is a vector space over ° 9 º » to the reader. For
)
part 2 , it is straightforward to show that : ² ³ is a submodule of 4 . Since
: ² ³ : and ; ² ³ ; we see that : ² ³ q ; ² ³ ~ ¸ ¹. Also, if # 4 ² ³ , then
# ~ . But #~ b ! for some : and !; and so ~ #~ b !.
Since : and ! ; we deduce that ~ ! ~ , whence # : ² ³ l ; ² ³ .
Thus, 4 ² ³ : ² ³ l ; ² ³ . But the reverse inequality is manifest.
(
Theorem 6.12 The cyclic decomposition theorem of a primary module) Let
4 be a primary finitely generated torsion module over a principal ideal domain
9 , with order .
)
1 4 is a direct sum
4 ~ ºº# »» l Ä l ºº# »» (6.1 )
of cyclic submodules with annihilators ann²ºº# »»³ ~ º » , which can be
arranged in ascending order
²
ann º# »»³ Ä ann ºº# »»³
²º
