Nil Clean Involutions

TL;DR

This paper proves that in rings where involutions are sums of idempotents and nilpotents, the idempotent must be 1, leading to a complete characterization of weakly nil-clean rings.

Contribution

It establishes a key property of involutions in rings and fully characterizes weakly nil-clean rings, advancing understanding of their algebraic structure.

Findings

Involutions as sums of idempotents and nilpotents have idempotent equal to 1.

Complete characterization of weakly nil-clean rings.

Clarifies the structure of rings with involutions.

Abstract

We prove that if an involution in a ring is the sum of an idempotent and a nilpotent then the idempotent in this decomposition must be 1. As a consequence, we completely characterize weakly nil-clean rings introduced recently in [Breaz, Danchev and Zhou, Rings in which every element is either a sum or a difference of a nilpotent and an idempotent, J. Algebra Appl., DOI: 10.1142/S0219498816501486].