Page 177 - Advanced Linear Algebra

Modules Over a Principal Ideal Domain 161

)
a Prove that a submodule of 4 is complemented if and only if 4 ° 5
5
is free.
b If 4 ) is also finitely generated, prove that 5 is complemented if and
only if 4°5 is torsion-free.
15. Let 4 be a free module of finite rank over a principal ideal domain .
9
a Prove that if 5 ) is a complemented submodule of 4 , then
rk²5³ ~ rk²4³ if and only if 5 ~ 4.
b Show that this need not hold if is not complemented.
)
5
c Prove that 5 ) is complemented if and only if any basis for 5 can be
extended to a basis for 4 .
16. Let 4 and be free modules of finite rank over a principal ideal domain
5
9 ¢. Let 4 ¦ 5 9 be an -homomorphism.
a Prove that ker²³ is complemented.
)

b) What about im²³ ?

c) Prove that
4
²
rk²4³~ rk²ker ² ³³ b rk im² ³³~ rk²ker ² ³³ b rk 6 7

ker ²³

d If ) is surjective, then is an isomorphism if and only if
rk²4³ ~ rk²5³.
)
e If is a submodule of 4 and if 4 ° 3 is free, then
3
4
rk 6 7 ~ rk² 4 ³ c rk² 3 ³
3
17. A submodule 5 of a module 4 is said to be pure in 4 if whenever
#¤4 ± 5, then #¤5 for all nonzero 9.
a Show that is pure if and only if  # 5 and ~ # $ for  9 implies
)
5
$5.
b Show that is pure if and only if 4 ° 5 is torsion-free.
)
5
c If is a principal ideal domain and 4 is finitely generated, prove that
)
9
5 4 is pure if and only if ° 5 is free.
d If and are pure submodules of 4 , then so are q 3 5 and r 3 5 .
)
5
3
What about 3b5 ?
e If 5 ) is pure in 4 , then show that 3 q 5 is pure in for any
3
submodule of 4 .
3
18. Let 4 be a free module of finite rank over a principal ideal domain . Let
9
3 5 and be submodules of 4 3 with complemented in 4 . Prove that
rk²3 b 5³ b rk²3 q 5³ ~ rk²3³ b rk²5³