College Algebra Exam Review 300

College Algebra Exam Review 300 - 6.6.7. Show that R D Z C...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
310 6. RINGS root r=s , where r and s are relatively prime, then s divides a n and r divides a 0 . In particular, if the polynomial is monic, then its only rational roots are integers. 6.6.4. Generalize the previous exercise to polynomials over a unique fac- torization domain. 6.6.5. Complete the details of this alternative proof of Gauss’s Lemma: Let R be a UFD. For any irreducible p 2 R , consider the quotient map ± p W R ±! R=pR , and extend this to a homomorphism ± p W RŒxŁ ±! .R=pR/ŒxŁ , defined by ± p . P a i x i / D P i ± p .a i /x i , using Corollary 6.2.8 . (a) Show that a polynomial h.x/ is in the kernel of ± p if, and only if, p is a common divisor of the coefficients of h.x/ . (b) Show that f.x/ 2 RŒxŁ is primitive if, and only if, for all irre- ducible p , ± p .f.x// ¤ 0 . (c) Show that .R=pR/ŒxŁ is integral domain for all irreducible p . (d) Conclude that if f.x/ and g.x/ are primitive in RŒxŁ , then f.x/g.x/ is primitive as well. 6.6.6. Show that Z Œ p ± satisfies the ascending chain condition for prin- cipal ideals but has irreducible elements that are not prime.
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 6.6.7. Show that R D Z C x Q x does not satisfy the ascending chain condition for principal ideals. Show that irreducibles in R are prime. 6.7. Noetherian Rings This section can be skipped without loss of continuity. The rings Z x and Kx;y;z are not principal ideal domains. How-ever, we shall prove that they have the weaker property that every ideal is nitely generatedthat is, for every ideal I there is a nite set S such that I is the ideal generated by S . A condition equivalent to the nite generation property is the ascend-ing chain condition for ideals . Denition 6.7.1. A ring (not necessarily commutative, not necessarily with identity) satises the ascending chain condition (ACC) for ideals if every strictly increasing sequence of ideals has nite length. We denote the ideal generated by a subset S of a ring R by .S/ ....
View Full Document

This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.

Ask a homework question - tutors are online