Bazzoni's conjecture and almost Prüfer domains. Issue 7 (3rd July 2019)

• Record Type: Journal Article
• Authors: Chang, Gyu Whan; Hamdi, Haleh
• Is Part Of: Communications in algebra. Volume 47:Issue 7(2019)
• Page Start: 2931
• Page End: 2940
• Publication Date: 2019-07-03
• Subjects: Almost Prüfer domain -- almost PvMD -- LPI domain -- t-LPI domain -- finite character -- finite t-character; 13A15 -- 13G05 -- 13F05; Algebra -- Periodicals; 512.005
• DOI: 10.1080/00927872.2018.1543426
• Languages: English
• ISSNs: 0092-7872

Abstract

Abstract: An almost Prüfer domain D is an integral domain in which for any 0 ≠ a, b ∈ D, there is an integer n = n ( a, b ) ≥ 1 such that ( a n, b n ) 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.