On $v$--domains and star operations

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 $\ast$--Prüfer (respectively, $(\ast ,v)$--Prüfer) domain if $(F{F}^{-1}{)}^{\ast }=D$ (respectively, $(F^v{F}^{-1}{)}^{\ast }=D$) for all $F\in \f(D)$. We establish that $\ast$--Prüfer domains (and $(\ast ,v)$--Prüfer domains) for various star operations $\ast$ span a major portion of the known generalizations of Prüfer 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 Prüfer domain, as a model, and we show which statements of that theorem on Prüfer domains can be generalized in a natural way and proved for $\ast$--Prüfer domains, and which cannot be. We also show that in a $\ast$--Prüfer 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 Prüfer $v$--multiplication domains. We also link $D$ being $\ast$--Prüfer (or $(\ast ,v)$--Prüfer) with the group Inv${}^{\ast }(D)$ of $\ast$-invertible $\ast$-ideals (under $\ast$-multiplication) being lattice-ordered.