Unformatted text preview: Q W R=.p/ ²! End .M/ such that D Q ı ± , where ± W R ²! R=.p/ is the quotient map. Hence M is a vector space over the ﬁeld R=.p/ . The action of R=.p/ on M is given by .r C .p//x D Q .r C .p//.x/ D .r/.x/ D rx: Suppose that ' W M ²! M is a surjective R –module homomorphism. For x 2 M , p'.x/ D '.px/ D . Thus p M D p'.M/ D f g , and M is a also an R=.p/ –vector space. Moreover, '..r C .p//x/ D '.rx/ D r'.x/ D .r C .p//'.x/; so ' is R=.p/ –linear. n We are now ready for the proof of uniqueness in Theorem 8.5.2 . Proof of Uniqueness in Theorem 8.5.2 Suppose that M has two direct sum decompositions: M D A ˚ A 1 ˚ A 2 ˚ ´´´ ˚ A s ; where µ A is free, µ for i ¶ 1 , A i Š R=.a i / , and...
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 Fall '08
 EVERAGE
 Algebra, Vector Space, Abelian group, kernel

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