Almost GCD Domains of Finite t-Character

Let D be an integral domain. Two nonzero elements x, y D are v-coprime if (x) ∩ (y) = (xy). D is an almost-GCD domain (AGCD domain) if for every pair x, y D, there exists a natural number n = n(x, y) such that (xn) ∩ (yn) 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 M} is finite, if and only if the set S(x) := {y D; y nonunit, y divides xn 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 ≤ r. One of the various consequences of this result is that a GCD domain D is a semilocal Bézout 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.