On the ideal avoidance property

TL;DR

This paper explores the avoidance property in ideals and rings, establishing new generalizations, characterizations, and conditions under which avoidance is preserved or reflected, including for graded rings and overring structures.

Contribution

It introduces a general avoidance lemma, characterizes avoidance in direct products and overrings, and links avoidance with properties like being a PIR and preservation under flat epimorphisms.

Findings

Every idempotent ideal has avoidance.

Avoidance property is preserved under flat ring epimorphisms.

Certain avoidance rings are characterized as PIRs.

Abstract

In this article, we investigate the avoidance property of ideals and rings. Among the main results, a general version of the avoidance lemma is formulated. It is shown that every idempotent ideal (and hence every pure ideal) has avoidance. The avoidance property of arbitrary direct products of avoidance rings is characterized. It is shown that every overring of an avoidance domain is an avoidance domain. Next, we show that every avoidance $\mathbb{N}$-graded ring whose base subring is a finite field is a PIR. It is also proved that the avoidance property is preserved under flat ring epimorphisms. Dually, we formulate a notion of strong avoidance, and show that it is reflected by pure morphisms.