Two questions on domains in which locally principal ideals are invertible

Abstract

An LPI domain is a domain in which every locally principal ideal is invertible. In this paper, we give a negative answer to a question of Anderson–Zafrullah. We show that if [Formula: see text] is an LPI domain and [Formula: see text] is a multiplicatively closed set, then [Formula: see text] is not necessarily an LPI domain. Concerning a question of Bazzoni, we give a characterization of finite character for a finitely stable LPI-domain. We show that if [Formula: see text] is a finitely stable LPI-domain, then every nonzero nonunit element of [Formula: see text] is contained in only finitely many maximal ideals which are in [Formula: see text], and [Formula: see text] is of finite character if and only if each nonzero nonunit element is contained in only finitely many nonstable maximal ideals. A maximal ideal [Formula: see text] is in [Formula: see text] provided there is a finitely generated ideal [Formula: see text] such that [Formula: see text] is the only maximal ideal containing [Formula: see text].