HW4_selected_solns

# HW4_selected_solns - College of the Holy Cross, Fall 2009...

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College of the Holy Cross, Fall 2009 Math 351 – Homework 4 selected solutions Chapter 16, #10. To show that R [ x ] and S [ x ] are isomorphic, we need to deﬁne a mapping from R [ x ] to S [ x ] and show that it is a ring homomorphism, 1-1, and onto. What we know is that R and S are isomorphic, which means there exists a map φ : R S that is a ring homom., 1-1, and onto. Let’s deﬁne a map δ : R [ x ] S [ x ] that takes a polynomial r n x n + ··· + r 1 x + r 0 and maps it to φ ( r n ) x n + ··· + φ ( r 1 ) x + φ ( r 0 ). Note that this really gives a map R [ x ] S [ x ] since a typical element of R [ x ] is a polynomial with coeﬃcients in R , and we’ve said how to map that to a polynomial with coeﬃcients in S . Since R [ x ] is not a quotient ring, we don’t need to worry about δ being well-deﬁned. Let’s show that δ is a homomorphism. We have δ (( a n x n + ··· + a 1 x + a 0 ) + ( b m x n + ··· + b 1 x + b 0 )) = δ ( c s x s + ··· + c 1 x + c 0 ) , where
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## This document was uploaded on 11/03/2010.

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