Bazzoni’s conjecture and almost Prüfer domains

Abstract

An almost Prüfer domain D is an integral domain in which for any , there is an integer such that is invertible. Hence, Prüfer domains are almost Prüfer. An integral domain D is of finite character if each nonzero nonunit of D is contained in only finitely many maximal ideals of D. Bazzoni’s conjecture states that if every nonzero locally principal ideal of a Prüfer domain D is invertible, then D is of finite character. This conjecture was proved in [Holland, Martinez, McGovern and Tesemma, Bazzoni’s Conjecture, J. Algebra 320 (2008), 1764–1768]. In this paper, we show that Bazzoni’s conjecture is true for almost Prüfer domains, that is, we prove that if every locally finitely generated ideal of an almost Prüfer domain D is finitely generated, then D is of finite character. This result will be proved in a more general setting of almost Prüfer v-multiplication domains.