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Unformatted text preview: Q W R=.p/ ! End .M/ such that D Q , where W R ! R=.p/ is the quotient map. Hence M is a vector space over the eld 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 ! 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 A is free, for i 1 , A i R=.a i / , and...
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.
- Fall '08