Almost GCD domains of finite t-character

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 (AGC D 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 s how 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 so me 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 mutua lly v-coprime elements of S(x) has length less than or equal to r. One of t he various consequences of this result is that a GCD domain D is a semiloca l Bezout domain if and only if D does not contain an infinite sequence of m utually v-coprime nonunit elements. Then, we study integrally closed AGCD d omains of finite t-character of the type A + XB[X] and we construct example s of nonintegrally closed AGCD of finite t-character by local algebra techn iques. (C) 2001 Academic Press.