Unformatted text preview: a j are mutually commuting, 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 RŒx 1 ;:::;x n Ł ±! R such that ' a .r/ D '.r/ for r 2 R and ' a .x j / D a j for 1 ² j ² n . We have ' a . X I r I x I / D X I '.r I / a I ; where for a multiindex I D .i 1 ;i 2 ;:::;i n / , a I denotes a i 1 1 a i 2 2 ³³³ a i n n . Proof. The proof is essentially the same as that of the one variable substitution principle. n...
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 Fall '08
 EVERAGE
 Logic, Algebra, Ri, unital ring homomorphism

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