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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 nitely 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 nitely 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 1 , p k 1 M=p k M R=.p/ if p k divides a and p k 1 M=p k M D f g otherwise. (b) Let M be a nitely generated torsion R module, with a direct sum decomposition M D A 1 A 2 A s ; where for i 1 , A i R=.a i / , and...
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- Fall '08