Rings generated by idempotents and nilpotents

TL;DR

This paper characterizes rings where each element can be expressed as sums of specific idempotents, tripotents, and nilpotents, revealing their structure is determined by additive decompositions of their squares.

Contribution

It provides new characterizations of rings based on element decompositions involving idempotents, tripotents, and nilpotents, linking these to the additive structure of squares.

Findings

Rings where every element is sum of two commuting idempotents and a nilpotent are characterized.

Rings where every element is sum of two tripotents and a nilpotent are characterized.

Such rings are fully determined by the additive decompositions of their square elements.

Abstract

We present new characterizations of the rings in which every element is the sum of two idempotents and a nilpotent that commute, and the rings in which every element is the sum of two tripotents and a nilpotent that commute. We prove that such rings are completely determined by the additive decompositions of their square elements.