Idempotence and divisorialty in Pr\"ufer-like domains

TL;DR

This paper explores ideal-theoretic properties like idempotence and divisoriality in Pr"ufer-like domains using semistar operations and Nagata rings, extending known properties from classical Pr"ufer domains.

Contribution

It generalizes ideal-theoretic properties related to idempotence and divisoriality to Pr"ufer $ ext{star}$-multiplication domains via semistar operations and Nagata rings.

Findings

Ideal-theoretic properties hold in Pr"ufer domains

Properties extend to Pr"ufer $ ext{star}$-multiplication domains

Semistar Nagata ring is instrumental in the analysis

Abstract

Let $D$ be a Pr\"ufer $\star$-multiplication domain, where $\star$ is a semistar operation on $D$. We show that certain ideal-theoretic properties related to idempotence and divisoriality hold in Pr\"ufer domains, and we use the associated semistar Nagata ring of $D$ to show that the natural counterparts of these properties also hold in $D$.