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College Algebra Exam Review 262

# College Algebra Exam Review 262 - a j are mutually...

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272 6. RINGS Proposition 6.2.5. (Substitution principle) Suppose that R and R 0 are rings with multiplicative identity, with R commutative, and ' W R ! R 0 is a unital ring homomorphism. For each a 2 R 0 , there is a unique unital ring homomorphism ' a W ROExŁ ! R 0 such that ' a .r/ D '.r/ for r 2 R , and ' a .x/ D a . We have ' a . X i r i x i / D X i '.r i /a i : Proof. If ' a is to be a homomorphism, then it must satisfy ' a . X i r i x i / D X i '.r i /a i : Therefore, we define ' a by this formula. It is then straightforward to check that ' a is a ring homomorphism. n There is also a multivariable version of the substitution principle, which formalizes evaluation of polynomials of several variables. Suppose that R and R 0 are rings with multiplicative identity, with R commutative, and ' W R ! R 0 is a unital ring homomorphism. Given an n –tuple a D .a 1 ; a 2 ; : : : ; a n / of elements in R 0 , we would like to have a homomor- phism from ROEx 1 ; : : : ; x n Ł to R 0 extending ' and sending each x j to a j . This only makes sense, however, if the elements
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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 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 multi-index 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 substi-tution principle. n...
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