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.
- Fall '08