Unformatted text preview: J are linearly dependent over R , and that J is a not a free R –module. 8.5.6. Show that M D Q = Z is a torsion Z –module, that M is not ﬁnitely generated, and that ann .M/ D f g . 8.5.7. Let R be a principal ideal domain. The purpose of this exercise is to give another proof of the uniqueness of the invariant factor decomposition for ﬁnitely generated torsion R –modules. Let p be an irreducible of R . (a) Let a be a nonzero, nonunit element of R and consider M D R=.a/ . Show that for k ² 1 , p k ± 1 M=p k M Š R=.p/ if p k divides a and p k ± 1 M=p k M D f g otherwise. (b) Let M be a ﬁnitely generated torsion R –module, with a direct sum decomposition M D A 1 ˚ A 2 ˚ ³³³ ˚ A s ; where ´ for i ² 1 , A i Š R=.a i / , and...
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- Fall '08