Almost GCD domains of finite t-character

JOURNAL OF ALGEBRA

卷:245

Article

Dumitrescu, T ; Lequain, Y ; Mott, JL ; Zafrullah, M

关键词: Bezout domain;    Prufer v-multiplication domain;    GCD domain;    Krull domain;    t-ideal;    v-coprime elements;    group of divisibility;   

DOI  :  10.1006/jabr.2001.8917

来源: Elsevier

【 摘 要 】

Let D be an integral. domain. Two nonzero elements x, y is an element of D are v-coprime if (x) boolean AND (y) = (xy). D is an almost-GCD domain (AGCD domain) if for every pair x, y is an element of D, there exists a natural number n = n (x, y) such that (x(n)) boolean AND (y(n)) is principal. We show that if x is a nonzero nonunit element of an almost GCD domain D, then the set {M; M maximal t-ideal, x is an element of M} is finite, if and only if the set S(x) := {y is an element of D; y nonunit, y divides x(n) for some n} does not contain an infinite sequence of mutually v-coprime elements, if and only if there exists an integer r such that every sequence of mutually v-coprime elements of S(x) has length less than or equal to r. One of the various consequences of this result is that a GCD domain D is a semilocal Bezout domain if and only if D does not contain an infinite sequence of mutually v-coprime nonunit elements. Then, we study integrally closed AGCD domains of finite t-character of the type A + XB[X] and we construct examples of nonintegrally closed AGCD of finite t-character by local algebra techniques. (C) 2001 Academic Press.

【 授权许可】

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