Local properties of integral domains under extensions and pullback constructions

TL;DR

This paper investigates how local properties of integral domains are preserved or transferred through various algebraic extensions and constructions, such as polynomial rings, overrings, and pullbacks.

Contribution

It provides new insights into the transfer of local properties of integral domains under multiple algebraic extensions and pullback constructions.

Findings

Characterizes conditions for property transfer under flat overrings.

Analyzes the behavior of local properties in Nagata ideal transforms.

Examines the preservation of properties in polynomial and quotient extensions.

Abstract

For a property $\mathcal{X}$ of integral domains, an integral domain $D$ is said to be a {\it locally $\mathcal{X}$-domain} if $D_P$ has the property $\mathcal{X}$ for every prime ideal $P$ of $D$. In this paper, we study the transfer of local properties of integral domains under several extensions and constructions, including flat overrings, Nagata ideal transforms, polynomial rings and their quotient extensions, and pullback constructions.