College Algebra Exam Review 263

College Algebra Exam Review 263 - X i T i denes a...

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6.2. HOMOMORPHISMS AND IDEALS 273 Corollary 6.2.7. (Evaluation of polynomials) Consider the ring RŒxŁ of polynomials over a commutative ring R with multiplicative identity. For any a 2 R , there is a unique homomorphisms ev a W RŒxŁ ! R with the property that ev a .r/ D r for r 2 R and ev a .x/ D a . We have ev a . X i r i x i / D X i r i a i : We usually denote ev a .p/ by p.a/ . Corollary 6.2.8. (Extensions of homomorphisms to polynomial rings) If W R ! R 0 is a unital homomorphism of commutative rings with multi- plicative identity, then there is a unique homomorphism Q W RŒxŁ ! R 0 ŒxŁ that extends . Proof. Apply Proposition 6.2.5 with the following data: Take ' W R ! R 0 ŒxŁ to be the composition of W R ! R 0 with the inclusion of R 0 into R 0 ŒxŁ , and set a D x . By the proposition, there is a unique homomorphism from RŒxŁ to R 0 ŒxŁ extending ' , and sending x to x . The extension is given by the formula Q . X i s i x i / D X i .s i /x i : n Example 6.2.9. Let V be a vector space over K and let T 2 End K .V / . Then ' T W X i ± i x i 7!
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Unformatted text preview: X i T i denes a homomorphism from Kx to End K .V / . What does this mean, and how does it follow from Proposition 6.2.5 ? End K .V / is a vector space over K as well as a ring. The product of a scalar 2 K and a linear map S 2 End K .V / is dened by .S/.v/ D S.v/ for v 2 V . Let I denote the identity endomorphism of V dened by I.v/ D v . Then I.v/ D v for v 2 V . The map ' W K ! End K .V / given by 7! I is easily seen to be a unital ring homomorphism from K to End K .V / . By Proposition 6.2.5 , there is a unique homomorphism ' T W Kx ! End K .V / with ' T .x/ D T and ' T ./ D I . Moreover, ' T . P i i x i / D P i . i I/T i D P i i T i . We usually write p.T / for ' T .p/ ....
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