Semistar Invertibility on Integral Domains

Abstract

After the introduction in 1994, by Okabe and Matsuda, of the notion of semistar operation, many authors have investigated different aspects of this general and powerful concept. A natural development of the recent work in this area leads to investigate the concept of invertibility in the semistar setting. In this paper, we will show the existence of a “theoretical obstruction” for extending many results, proved for star-invertibility, to the semistar case. For this reason, we will introduce two distinct notions of invertibility in the semistar setting (called ⋆–invertibility and quasi–⋆–invertibility), we will discuss the motivations of these “two levels” of invertibility and we will extend, accordingly, many classical results proved for the d–, v–, t– and w– invertibility. Among the main properties proved here, we mention the following: (a) several characterizations of ⋆–invertibility and quasi–⋆–invertibility and necessary and sufficient conditions for the equivalence of these two notions; (b) the relations between the ⋆–invertibility (or quasi–⋆–invertibility) and the invertibility (or quasi–invertibility) with respect to the semistar operation of finite type (denoted by ⋆ f ) and to the stable semistar operation of finite type (denoted by ⋆̃), canonically associated to ⋆; (c) a characterization of the H(⋆)–domains in terms of semistar–invertibility (note that the H(⋆)–domains generalize, in the semistar setting, the H–domains introduced by Glaz and Vasconcelos); (d) for a semistar operation of finite type a nonzero finitely generated (fractional) ideal I is ⋆–invertible (or, equivalently, quasi–⋆–invertible, in the stable semistar case) if and only if its extension to the Nagata semistar ring I Na(D, ⋆) is an invertible ideal of Na(D, ⋆).

Cite this paper

@inproceedings{Fontana2004SemistarIO, title={Semistar Invertibility on Integral Domains}, author={Marco Fontana and GIAMPAOLO PICOZZA}, year={2004} }