Page 131 - Advanced Linear Algebra
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Modules I: Basic Properties 115
Torsion Elements
-
In a vector space over a field , singleton sets # ¸ ¹ where £ # are linearly
=
independent. Put another way, £ and # £ imply # £ . However, in a
module, this need not be the case.
Example 4.4 The abelian group { is a -module, with
{ ~ ¸ Á Á Ã Á c ¹
scalar multiplication defined by ' ~ ²' h ³ mod , for all ' { and { .
However, since ~ for all { , no singleton set ¸ ¹ is linearly
independent. Indeed, { has no linearly independent sets.
This example motivates the following definition.
9
Definition Let 4 be an -module. A nonzero element #4 for which #~
for some nonzero 9 is called a torsion element of 4 . A module that has no
nonzero torsion elements is said to be torsion-free . If all elements of 4 are
torsion elements, then 4 is a torsion module . The set of all torsion elements of
4 4, together with the zero element, is denoted by tor .
If 4 is a module over an integral domain , it is not hard to see that 4 tor is a
(
submodule of 4 and that 4 ° 4 tor is torsion-free. We will define quotient
modules shortly: they are defined in the same way as for vector spaces.)
Annihilators
Closely associated with the notion of a torsion element is that of an annihilator.
Definition Let 4 be an -module. The annihilator of an element # 4 is
9
ann²#³ ~¸ 9 # ~ ¹
and the annihilator of a submodule of 4 is
5
ann²5³ ~ ¸ 9 5 ~ ¸ ¹¹
where 5 ~ ¸ # # 5¹ . Annihilators are also called order ideals .
9
It is easy to see that ann²#³ and ann²5³ are ideals of . Clearly, # 4 is a
torsion element if and only if ann²#³ £ ¸ ¹ . Also, if and are submodules of
(
)
4, then
( ) ¬ ann ²)³ ann ²(³
(note the reversal of order).
Let 4~ ºº" Á à Á " »» be a finitely generated module over an integral domain
9 " and assume that each of the generators is torsion, that is, for each , there is
a nonzero ann ²" ³ . Then, the nonzero product ~ Ä annihilates each
generator of 4 and therefore every element of 4 , that is, ann ² 4 ³ . This
