Generalizing Semi-$n$-Potent Rings

TL;DR

This paper introduces and characterizes a new class of rings where each element can be expressed as a sum of a tripotent and a commuting element from a specific subring, generalizing previous results.

Contribution

It defines the class of semi-$n$-potent rings and provides a complete description of their structure modulo the Jacobson radical, extending earlier work by Koşan-Yildirim-Zhou.

Findings

Rings are characterized as direct products of Boolean and Yaqub rings.

Complete structural description of these rings modulo their Jacobson radical.

Generalization of previous results on ring decompositions.

Abstract

We define and explore the class of rings $R$ for which each element in $R$ is a sum of a tripotent element from $R$ and an element from the subring $\Delta(R)$ of $R$ which commute each other. Succeeding to obtain a complete description of these rings modulo their Jacobson radical as the direct product of a Boolean ring and a Yaqub ring, our results somewhat generalize those established by Ko\c{s}an-Yildirim-Zhou in Can. Math. Bull. (2019).