Page 154 - Advanced Linear Algebra

138 Advanced Linear Algebra

I
of 4 has a maximal element. That is, for every nonempty collection of
submodules of 4 there is an :  I with the property that
; I ¬ ;‹ :.
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7. Let ¢4 ¦ 5 be an -homomorphism.
a Show that if 4 is finitely generated, then so is im ³ ² .
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b Show that if ker²³ and im²³ are finitely generated, then

4~ ker ² ³ b : where : is a finitely generated submodule of 4.

Hence, 4 is finitely generated.
8. If is Noetherian and is an ideal of show that ° ? 9 9 ? is also Noetherian.
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9. Prove that if is Noetherian, then so is 9´% ÁÃÁ% µ .
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10. Find an example of a commutative ring with identity that does not satisfy
the ascending chain condition.
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11. a Prove that an -module 4 is cyclic if and only if it is isomorphic to
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9° where is an ideal of 9.
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b Prove that an -module 4 is simple (4 £ ¸ ¹ and 4 has no proper
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)
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nonzero submodules if and only if it is isomorphic to 9° ? where is
a maximal ideal of .
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c Prove that for any nonzero commutative ring with identity, a simple
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9-module exists.
12. Prove that the condition that be a principal ideal domain in part 2 of
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Theorem 5.8 is required.
13. Prove Theorem 5.8 in the following way.
a Show that if ;‹ : are submodules of 4 and if and :°; are
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;
finitely generated, then so is .
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b The proof is again by induction. Assuming it is true for any module
generated by elements, let 4 ~ ºº# ÁÃÁ# b »» and let
Z
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Z
4~ ºº# Á à Á # »». Then let ; ~ : q 4 in part a .

14. Prove that any -module 4 is isomorphic to the quotient of a free module
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- 4. If is finitely generated, - then can also be taken to be finitely
generated.
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:
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15 Prove that if and are isomorphic submodules of a module 4 it does
not necessarily follow that the quotient modules 4°: and 4°; are
as modules it does not
isomorphic. Prove also that if :l ; š :l ;
. Prove that these statements do hold if all
necessarily follow that ;š ;
modules are free and have finite rank.