Page 162 - Advanced Linear Algebra
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146 Advanced Linear Algebra
# b " b Ä b " ~
, then
If ~ Ä c
4 ~ ºº" Á ÃÁ" Á# ÁÃÁ# c »» ºº" ÁÃÁ" »»
and since the latter is a free module, so is 4 , and therefore so is 4 .
The Primary Cyclic Decomposition Theorem
The first step in the decomposition of a finitely generated module 4 over a
principal ideal domain is an easy one.
9
Theorem 6.9 Any finitely generated module 4 over a principal ideal domain 9
is the direct sum of a finitely generated free -module and a finitely generated
9
torsion -module
9
4~ 4 free l 4 tor
The torsion part 4 tor is unique, since it must be the set of all torsion elements of
4 4, whereas the free part free is unique only up to isomorphism, that is, the
rank of the free part is unique.
Proof. It is easy to see that the set 4 tor of all torsion elements is a submodule of
4 4 and the quotient ° 4 tor is torsion-free. Moreover, since 4 is finitely
generated, so is 4°4 tor . Hence, Theorem 6.8 implies that 4°4 tor is free.
Hence, Theorem 5.6 implies that
4~ 4 tor l -
where - 4°4 tor is free.
;
As to the uniqueness of the torsion part, suppose that 4~ ; l . where is
torsion and is free. Then ; 4 . But if # ~ ! tor b 4 tor for ! ; and
.
., then ~ # c ! 4 and so ~ and # ; . Thus, ; ~ 4 .
tor
tor
-
For the free part, since 4 ~ 4 l tor - ~ 4 l tor . , the submodules and .
are both complements of 4 tor and hence are isomorphic.
Note that if ¸$ ÁÃÁ$ ¹ is a basis for 4 free we can write
4 ~ ºº$ »» l Ä l ºº$ »» l 4 tor
where each cyclic submodule ºº$ »» has zero annihilator. This is a partial
decomposition of 4 into a direct sum of cyclic submodules.
The Primary Decomposition
In view of Theorem 6.9, we turn our attention to the decomposition of finitely
generated torsion modules 4 over a principal ideal domain. The first step is to
decompose 4 into a direct sum of primary submodules, defined as follows.
