On $v$--domains and star operations

TL;DR

This paper explores the properties of $ ext{star}$-Pr"ufer domains, generalizing classical Pr"ufer domain concepts through star operations, and investigates their algebraic and order-theoretic structures.

Contribution

It introduces and analyzes $ ext{star}$-Pr"ufer domains, extending known Pr"ufer domain properties to broader classes via star operations and linking ideal invertibility with lattice structures.

Findings

$ ext{star}$-Pr"ufer domains encompass many known generalizations of Pr"ufer domains.

In $ ext{star}$-Pr"ufer domains, pairs of $ ext{star}$-invertible ideals admit GCDs.

The group of $ ext{star}$-invertible ideals is lattice-ordered.

Abstract

Let $\ast$ be a star operation on an integral domain $D$. Let $\f(D)$ be the set of all nonzero finitely generated fractional ideals of $D$. Call $D$ a $\ast$--Pr\"ufer (respectively, $(\ast, v)$--Pr\"ufer) domain if $(FF^{-1})^{\ast}=D$ (respectively, $(F^vF^{-1})^{\ast}=D$) for all $F\in \f(D)$. We establish that $\ast$--Pr\"ufer domains (and $(\ast, v)$--Pr\"ufer domains) for various star operations $\ast $ span a major portion of the known generalizations of Pr\"{u}fer domains inside the class of $v$--domains. We also use Theorem 6.6 of the Larsen and McCarthy book [Multiplicative Theory of Ideals, Academic Press, New York--London, 1971], which gives several equivalent conditions for an integral domain to be a Pr\"ufer domain, as a model, and we show which statements of that theorem on Pr\"ufer domains can be generalized in a natural way and proved for $\ast$--Pr\"ufer domains, and which cannot be. We also show that in a $\ast $--Pr\"ufer domain, each pair of $\ast $-invertible $\ast $-ideals admits a GCD in the set of $\ast $-invertible $\ast $-ideals, obtaining a remarkable generalization of a property holding for the "classical" class of Pr\"ufer $v$--multiplication domains. We also link $D$ being $\ast $--Pr\"ufer (or $(\ast, v)$--Pr\"ufer) with the group Inv$^{\ast}(D)$ of $\ast $-invertible $\ast $-ideals (under $\ast$-multiplication) being lattice-ordered.