Page 176 - Advanced Linear Algebra

160 Advanced Linear Algebra

c) 4 is cyclic of prime power order.
3. Let be a principal ideal domain and 9 b the field of quotients. Then 9 b is
9
an -module. Prove that any nonzero finitely generated submodule of 9 b
9
is a free module of rank .

4. Let be a principal ideal domain. Let 4 be a finitely generated torsion-
9
free -module. Suppose that is a submodule of 4 for which is a free
9
5
5
9 -module of rank and 4 ° 5 is a torsion module. Prove that 4 is a free
9 -module of rank .
5. Show that the primary cyclic decomposition of a torsion module over a
principal ideal domain is not unique even though the elementary divisors
(
are .
)
9
6. Show that if 4 is a finitely generated -module where is a principal
9
ideal domain, then the free summand in the decomposition 4~ - l 4 tor
need not be unique.
9

7. If ºº#»» is a cyclic -module of order show that the map ¢ 9 ¦ ºº#»»
9
defined by ~ # is a surjective -homomorphism with kernel º » and so
9
ºº#»» š
º »
8. If is an integral domain with the property that all submodules of cyclic
9
9 9-modules are cyclic, show that is a principal ideal domain.
9. Suppose that is a finite field and let - i be the set of all nonzero elements
-
of .
-
)
a Show that if ²%³  -´%µ is a nonconstant polynomial over and if
-
 - is a root of ²%³, then % c is a factor of ²%³.
)
b Prove that a nonconstant polynomial ²%³  -´%µ of degree can have

at most distinct roots in .

-
)
c Use the invariant factor or primary cyclic decomposition of a finite -
{
module to prove that - i is cyclic.
10. Let be a principal ideal domain. Let 4 ~ º º # » » be a cyclic -module
9
9
with order . We have seen that any submodule of 4 is cyclic. Prove that

for each 9 such that “ there is a unique submodule of 4 of order

.
11. Suppose that 4 is a free module of finite rank over a principal ideal
9
domain . Let 5 be a submodule of 4 . If 4 ° 5 is torsion, prove that
rk²5³ ~ rk²4³.
Z
-
12. Let -´%µ be the ring of polynomials over a field and let - ´%µ be the ring
%
of all polynomials in -´%µ that have coefficient of equal to . Then -´%µ

Z
is an - ´%µ -module. Show that -´%µ is finitely generated and torsion-free
Z
but not free. Is -´%µ a principal ideal domain?
13. Show that the rational numbers form a torsion-free -module that is not
r
{
free.
More on Complemented Submodules
14. Let be a principal ideal domain and let 4 be a free -module.
9
9