College Algebra Exam Review 266

College Algebra Exam Review 266 - 276 6. RINGS Example...

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Unformatted text preview: 276 6. RINGS Example 6.2.17. The kernel of the ring homomorphism KŒx ! K given by p 7! p.a/ is the set of all polynomials p having a as a root. Example 6.2.18. The kernel of the ring homomorphism C.R/ ! C.S / given by f 7! fjS is the set of all continuous functions whose restriction to S is zero. Example 6.2.19. Let K be a field. Define a map ' from KŒx to Fun.K; K/, the ring of K –valued functions on K by '.p/.a/ D p.a/. (That is, '.p/ is the polynomial function on K corresponding to the polynomial p .) Then ' is a ring homomorphism. The homomorphism property of ' follows from the homomorphism property of p 7! p.a/ for a 2 K . Thus '.p C q/.a/ D .p C q/.a/ D p.a/ C q.a/ D '.p/.a/ C '.q/.a/ D .'.p/ C '.q//.a/, and similarly for multiplication. The kernel of ' is the set of polynomials p such that p.a/ D 0 for all a 2 K . If K is infinite, then the kernel is f0g, since no nonzero polynomial with coefficients in a field has infinitely many roots. If K is finite, then ' is never injective. That is, there always exist nonzero polynomials p 2 KŒx such that p.a/ D 0 for all a 2 K . Indeed, Q we need merely take p.x/ D a2K .x a/. Definition 6.2.20. A ring R with no ideals other than f0g and R itself is said to be simple. Any field is a simple ring. You are asked to verify this in Exercise 6.2.10. In Exercise 6.2.11, you are asked to show that the ring M of n-by-n matrices with real entries is simple. This holds equally well for matrix rings over any field. Proposition 6.2.21. (a) Let fI˛ g be any collection of ideals in a ring R. Then \ I˛ is ˛ (b) an ideal of R. Let In be an increasing sequence of ideals in a ring R. Then [ In is an ideal of R. n Proof. Part (a) is an Exercise 6.2.17. For part (b), let x; y 2 I D [ In . n Then there exist k; ` 2 N such that x 2 Ik and y 2 I` . If n D maxfk; `g, ...
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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.

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