Extensions of I-Reversible Rings

TL;DR

This paper explores the properties of i-reversible rings, providing new examples, characterizations, and conditions for various ring extensions and matrix rings, expanding understanding of their structure.

Contribution

It introduces non-trivial i-reversible subrings of matrix rings, characterizes maximal i-reversible subrings of upper triangular matrices, and establishes conditions for i-reversibility in extensions and polynomial rings.

Findings

Constructed non-trivial i-reversible subrings of matrix rings for n ≥ 3.

Identified maximal i-reversible subrings of upper triangular matrix rings over fields.

Provided sufficient conditions for i-reversibility in polynomial and skew polynomial rings.

Abstract

A ring $R$ is said to be i-reversible if for every $a,b$ $\in$ $R$, $ab$ is a non-zero idempotent implies $ba$ is an idempotent. It is known that the rings $M_n(R)$ and $T_n(R)$ (the ring of all upper triangular matrices over $R$) are not i-reversible for $n \geq 3$. In this article, we provide a non-trivial i-reversible subring of $M_n(R)$ when $n \geq 3$ and $R$ has only trivial idempotents. We further provide a maximal i-reversible subring of $T_n(R)$ for each $n\geq 3$, if $R$ is a field. We then give conditions for i-reversibility of Trivial, Dorroh and Nagata extensions. Finally, we give some independent sufficient conditions for i-reversibility of polynomial rings, and more generally, of skew polynomial rings.