Duals and Transforms of Ideals in PVMDs

TL;DR

This paper investigates the properties of duals and transforms of t-ideals in PVMDs, focusing on when they form rings, their endomorphism rings, and related overrings, extending classical results from Prüfer domains.

Contribution

It introduces new conditions under which the dual of a t-ideal in PVMDs is a ring and explores the structure of overrings like Nagata transforms and endomorphism rings.

Findings

Characterization of when duals of t-ideals are rings

Conditions for duals to coincide with endomorphism rings

Analysis of overrings such as Nagata transforms in PVMDs

Abstract

In this paper we study when the dual of a $t$-ideal in a $PVMD$ is a ring? and we treat the question when it coincides with its endomorphism ring. We also study particular classes of overrings of PVMDs. Specially, we investigate the Nagata transform and the endomorphism ring of ideals in PVMDs in an attempt to establish analogues for well-known results on overrings of Pr\"ufer domains.