Well-centered Overrings of a Commutative Ring in Pullbacks and Trivial Extensions

TL;DR

This paper explores the concept of well-centered overrings in commutative rings, extending previous notions from integral domains and analyzing their transfer properties in pullbacks and trivial extensions to identify new classes of rings with this property.

Contribution

It generalizes the notion of well-centered overrings to all commutative rings and investigates their behavior in pullbacks and trivial extensions, addressing open questions in the field.

Findings

Identifies new classes of commutative rings with well-centered overrings.

Analyzes transfer properties of well-centered overrings in various ring extensions.

Provides insights into open problems posed by Heinzer and Roitman.

Abstract

Let $R$ be a commutative ring with identity and $T(R)$ its total quotient ring. We extend the notion of well-centered overring of an integral domain to an arbitrary commutative ring and we investigate the transfer of this property to different extensions of commutative rings in both integral and non-integral cases. Namely in pullbacks and trivial extensions. Our aim is to provide new classes of commutative rings satisfying this property and to shed light on some open questions raised by Heinzer and Roitman in \cite{HR}.