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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. However, 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 ascending 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/ ....
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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
 EVERAGE
 Algebra, Polynomials, Integers

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