College Algebra Exam Review 302

College Algebra Exam Review 302 - a D P j r j d.n j Then P...

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312 6. RINGS that deg .f ± g/ < deg .g/ . Then f ± g 2 J 0 , by the induction hypothesis, so f D g C .f ± g/ 2 J 0 . n Theorem 6.7.5. (Hilbert’s basis theorem). Suppose R is a commutative Noetherian ring with identity element. Then RŒxŁ is Noetherian. Proof. Let J be an ideal in RŒxŁ . Let m denote the minimum degree of nonzero elements of J . For each k ² 0 , let A k denote f a 2 R W a D 0 or there exists f 2 J with leading term ax k g : It is easy to check that A k is an ideal in R and A k ³ A k C 1 for all k . Let A D [ k A k . Since every ideal in R is finitely generated, there is a natural number N such that A D A N . For m ´ k ´ N , let f d .k/ j W 1 ´ j ´ s.k/ g be a finite generating set for A k , and for each j , let g .k/ j be a polynomial in J with leading term d .k/ j x k . Let J 0 be the ideal generated by f g .k/ j W m ´ k ´ N; 1 ´ j ´ s.k/ g . We claim that J D J 0 . Let f 2 J have degree n and leading term ax n . If n ² N , then a 2 A n D A N , so there exist r j 2 R such that a D P j r j d .N/ j . Then P j r j x n ± N g .N/ j is an element of J 0 with leading term ax n , so deg .f ± g/ < deg .f / . If m ´ n < N , then a 2 A n , so there exist r j 2 R such that
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Unformatted text preview: a D P j r j d .n/ j . Then P j r j g .n/ j is an element of J with lead-ing term ax n , so deg .f ± g/ < deg .f / . Thus for all nonzero f 2 J , there exists g 2 J such that deg .f ± g/ < deg .f / . It follows from Lemma 6.7.4 that J D J . n Corollary 6.7.6. If R is a commutative Noetherian ring, then RŒx 1 ;:::;x n Ł is Noetherian for all n . In particular, Z Œx 1 ;:::;x n Ł and, for all fields K , KŒx 1 ;:::;x n Ł are Noetherian. Proof. This follows from the Hilbert basis theorem and induction. n Exercises 6.7 6.7.1. Let R be a commutative ring with identity element. Let J be an ideal in RŒxŁ . Show that for each k ² , the set A k D f a 2 R W a D or there exists f 2 J with leading term ax k 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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