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Unformatted text preview: a j are mutually commut-ing, that is, a i a j D a j a i for every i and j . Proposition 6.2.6. (Multivariable subsitution principle) Suppose that R and R are rings with multiplicative identity, with R commutative, and ' W R ! R is a unital ring homomorphism. Given an n tuple a D .a 1 ;a 2 ;:::;a n / of mutually commuting elements in R there is a unique unital ring homomorphism ' a W Rx 1 ;:::;x n ! R such that ' a .r/ D '.r/ for r 2 R and ' a .x j / D a j for 1 j n . We have ' a . X I r I x I / D X I '.r I / a I ; where for a multi-index I D .i 1 ;i 2 ;:::;i n / , a I denotes a i 1 1 a i 2 2 a i n n . Proof. The proof is essentially the same as that of the one variable substi-tution principle. n...
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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