Semistar Invertibility on Integral Domains

Abstract

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, ⋆).

Prüfer ⋆–multiplication domains and ⋆–coherence

“…, I is f -invertible if and only if I isinvertible and F −1 is f -finite [17,Proposition 2.6]. Therefore this equivalence follows easily from Proposition 12 ((ii)⇔(i)) and Proposition 2 (3).…”

“…In particular, if I is f -finite, then it is -finite. The converse is not true and it is possible to prove that I is f -finite if and only if there exists J ∈ f (D), J ⊆ I, such that J = I [17,Lemma 2.3].…”

“…We denote by Inv(D, ) the set of all the -invertible ideals of D. From the definitions, it follows easily that an ideal is -invertible if and only if it is -invertible (and so I is˜ -invertible if and only if I is f -invertible). If I is f -invertible, then I and I −1 are f -finite [17,Proposition 2.6].…”

A Note on Prüfer Semistar Multiplication Domains

“…In [7], we have introduced the notion of quasi--invertibility, as a generalization, "typical" of the semistar context, of the notion of -invertibility:…”

“…To define P MD's they used a notion of -invertibility analogous to the one already used for star operations. In [7] we have introduced the notion of quasi semistar invertibility, which is more natural in the semistar context. In this note we want to give a new generalization of the notions of Prüfer domain and PvMD which uses quasi semistar invertibility, the "quasi P MD", and compare them with the P MD.…”