Unformatted text preview: s D t by uniqueness of dimension. We assume now that the length of m is greater than one and that the uniqueness assertion holds for all ﬁnitely generated torsion modules with a period of smaller length. Let p be an irreducible in R . Then x 7! px is a module endomorphism of M that maps each A i into itself. According to Lemma 8.5.6 , if p divides a i then A i =pA i Š R=.p/ , but if p is relatively prime to a i , then A i =pA i D f g . We have M=pM Š .A 1 ˚ A 2 ˚ ³³³ ˚ A s /=.pA 1 ˚ pA 2 ˚ ³³³ ˚ pA s / Š A 1 =pA 1 ˚ A 2 =pA 2 ˚ ³³³ ˚ A s =pA s Š .R=.p// k ; where k is the number of a i such that p divides a i . Since p.M=pM/ D f g , according to Lemma 8.5.7 , all the R –modules in view here are actually R=.p/ –vector spaces and the isomorphisms are R=.p/ –linear. It follows that the number k is the dimension of M=pM...
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 Fall '08
 EVERAGE
 Algebra, Vector Space, Prime number, Lemma, Ai =pAi

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