Two questions on domains in which locally principal ideals are invertible

摘要

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 R is an LPI domain and S is a multiplicatively closed set, then R-S 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 R is a finitely stable LPI-domain, then every nonzero nonunit element of R is contained in only finitely many maximal ideals which are in T (R), and R is of finite character if and only if each nonzero nonunit element is contained in only finitely many nonstable maximal ideals. A maximal ideal m is in T (R) provided there is a finitely generated ideal I such that m is the only maximal ideal containing I.