Rings with the Beachy-Blair condition

TL;DR

This paper explores the properties and relationships of rings satisfying the Beachy-Blair condition, especially in comparison with zip rings, including their behavior under certain extensions and asymmetries.

Contribution

It clarifies the distinctions between zip rings and rings with the Beachy-Blair condition and examines their behavior under skew polynomial and power series extensions.

Findings

Not all zip rings are symmetric in zip property

Examples of rings satisfying Beachy-Blair but not zip

Behavior of these properties under ring extensions

Abstract

A ring satisfies the left Beachy-Blair condition if each of its faithful left ideal is cofaithful. Every left zip ring satisfies the left Beachy-Blair condition, but both properties are not equivalent. In this paper we will study the similarities and the differences between zip rings and rings with the Beachy-Blair condition. We will also study the relationship between the Beachy-Blair condition of a ring and its skew polynomial and skew power series extensions. We give an example of a right zip ring that is not left zip, proving that the zip property is not symmetric.