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. However, 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 ascending 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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 Fall '08
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 Algebra, Polynomials, Integers, Integral domain, Commutative ring, Principal ideal domain, chain condition, principal ideals

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