Page 142 - Advanced Linear Algebra
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126 Advanced Linear Algebra
12. Prove that if a nonzero commutative ring with identity has the property
9
that every finitely generated -module is free then is a field.
9
9
;
13. Let 4 and 5 be 9 -modules. If is a submodule of 4 and is a
:
submodule of show that
5
4l 5 4 5
^
:l ; : ;
14. If is a commutative ring with identity and is an ideal of , then is an
9
?
9
?
9-module. What is the maximum size of a linearly independent set in ?
?
Under what conditions is free?
?
)
15. a Show that for any module 4 over an integral domain the set 4 tor of all
torsion elements in a module 4 is a submodule of 4 .
b Find an example of a ring with the property that for some -module
)
9
9
4 4 the set tor is not a submodule.
c ) Show that for any module 4 over an integral domain, the quotient
is torsion-free.
module 4°4 tor
)
16. a Find a module 4 that is finitely generated by torsion elements but for
which ann²4³ ~ ¸ ¹ .
b Find a torsion module 4 for which ann²4³ ~ ¸ ¹ .
)
17. Let be an abelian group together with a scalar multiplication over a ring
5
9 9 that satisfies all of the properties of an -module except that # does not
#
necessarily equal for all # 5 . Show that 5 can be written as a direct
sum of an -module 5 and another “pseudo -module” 5 .
9
9
9
18. Prove that hom 9 ²4Á 5³ is an -module under addition of functions and
scalar multiplication defined by
² ³²#³ ~ ² #³ ~ ² #³
19. Prove that any -module 4 is isomorphic to the -module hom 9 ² 9 Á 4 . ³
9
9
20. Let and be commutative rings with identity and let ¢ 9 ¦ : be a ring
9
:
homomorphism. Show that any -module is also an -module under the
:
9
scalar multiplication
# ~ ² ³#
21. Prove that hom { ² Á{{ ³ { where ~ gcd² Á . ³
22. Suppose that is a commutative ring with identity. If and are ideals of
@
9
?
9 9 for which ° 9 ° @ as -modules, then prove that ? ~ @ . Is the
9 ?
?
result true if 9° 9° @ as rings?
