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College Algebra Exam Review 300

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

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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 ROExŁ ! .R=pR/OExŁ , 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 ROExŁ is primitive if, and only if, for all irre- ducible p , p .f .x// ¤ 0 . (c) Show that .R=pR/OExŁ is integral domain for all irreducible p . (d) Conclude that if f .x/ and g.x/ are primitive in ROExŁ , then f .x/g.x/ is primitive as well. 6.6.6. Show that Z OE p satisfies the ascending chain condition for prin- cipal ideals but has irreducible elements that are not prime.
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Unformatted text preview: 6.6.7. Show that R D Z C x Q ŒxŁ 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 ŒxŁ and KŒx;y;zŁ are not principal ideal domains. How-ever, we shall prove that they have the weaker property that every ideal is finitely generated—that is, for every ideal I there is a finite set S such that I is the ideal generated by S . A condition equivalent to the finite generation property is the ascend-ing chain condition for ideals . Definition 6.7.1. A ring (not necessarily commutative, not necessarily with identity) satisfies the ascending chain condition (ACC) for ideals if every strictly increasing sequence of ideals has finite length. We denote the ideal generated by a subset S of a ring R by .S/ ....
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