Page 161 - Advanced Linear Algebra

Modules Over a Principal Ideal Domain 145

rk²4³ ~ rk²ker ² ³³ b rk²4³
and so rk²ker ² ³³ ~ , that is, ker ² ³ ~¸ ¹ , which implies that is an -

9
isomorphism and so is a basis.
:
In general, a basis for a submodule of a free module over a principal ideal
domain cannot be extended to a basis for the entire module. For example, the set
¸ ¹ is a basis for the submodule of the -module , but this set cannot be
{
{
{
extended to a basis for itself. We state without proof the following result
{
along these lines.
Theorem 6.7 Let 4 be a free -module of rank , where is a principal ideal
9

9
domain. Let be a submodule of 4 that is free of rank  . Then there is a
5
basis 8 for 4 that contains a subset : ~ ¸ # Á Ã Á # ¹ for which
¸ # Á ÃÁ # ¹ is a basis for , for some nonzero elements ÁÃÁ of 9.
5

Torsion-Free and Free Modules
Let us explore the relationship between the concepts of torsion-free and free. It
is not hard to see that any free module over an integral domain is torsion-free.
The converse does not hold, unless we strengthen the hypotheses by requiring
that the module be finitely generated.
Theorem 6.8 A finitely generated module over a principal ideal domain is free
if and only if it is torsion-free.
Proof. We leave proof that a free module over an integral domain is torsion-free
to the reader. Let . ~ ¸# ÁÃÁ# ¹ be a generating set for 4 . Consider first the

.
case ~ , whence . ~¸#¹ . Then is a basis for 4 since singleton sets are
linearly independent in a torsion-free module. Hence, 4 is free.
Now suppose that . ~ ¸"Á#¹ is a generating set with "Á# £ . If is linearly
.
independent, we are done. If not, then there exist nonzero Á  9 for which
" ~ #. It follows that 4 ~ ºº"Á #»» ‹ºº"»» and so 4 is a submodule of a
free module and is therefore free by Theorem 6.5. But the map ¢4 ¦ 4
defined by #~ # is an isomorphism because 4 is torsion-free. Thus 4 is
also free.
Now we can do the general case. Write

. ~ ¸" ÁÃÁ" Á# ÁÃÁ# c ¹
(
.
where : ~ ¸" ÁÃÁ" ¹ is a maximal linearly independent subset of . Note

that is nonempty because singleton sets are linearly independent.)
:
For each , the set ¸ # " Á Ã Á " Á # ¹ is linearly dependent and so there exist
 9 and ÁÃÁ  9 for which