Reversible ring property via idempotent elements

TL;DR

This paper introduces and studies the concept of right and left e-reversible rings, exploring their properties and relationships with other ring classes, revealing asymmetries and characterizations involving idempotent elements.

Contribution

It defines right and left e-reversible rings, analyzes their properties, and establishes their relationships with other ring classes, highlighting asymmetries and characterizations.

Findings

Right e-reversible rings are not symmetric with respect to left and right.

In semiprime rings, right e-reversibility coincides with being e-reduced, e-symmetric, and right e-semicommutative.

In right e-reversible rings, primeness is equivalent to being a domain.

Abstract

Regarding the question of how idempotent elements affect reversible property of rings, we study a version of reversibility depending on idempotents. In this perspective, we introduce {\it right} (resp., {\it left}) {\it $e$-reversible rings}. We show that this concept is not left-right symmetric. Basic properties of right $e$-reversibility in a ring are provided. Among others it is proved that if $R$ is a semiprime ring, then $R$ is right $e$-reversible if and only if it is right $e$-reduced if and only if it is $e$-symmetric if and only if it is right $e$-semicommutative. Also, for a right $e$-reversible ring $R$, $R$ is a prime ring if and only if it is a domain. It is shown that the class of right $e$-reversible rings is strictly between that of $e$-symmetric rings and right $e$-semicommutative rings.