Uniquely weakly nil-clean conditions on zero-divisors

TL;DR

This paper characterizes rings where every zero-divisor can be uniquely expressed as a sum or difference of a nilpotent and an idempotent, revealing structural conditions like being a D-ring or having a Boolean quotient.

Contribution

It provides a complete classification of rings with zero-divisors that are uniquely weakly nil-clean, extending the understanding of their algebraic structure and properties.

Findings

Rings where every zero-divisor is uniquely weakly nil-clean are characterized as D-rings or certain abelian, periodic rings.

The structure of such rings' quotients is described as isomorphic to fields, Boolean rings, or specific direct sums.

Conditions for zero-divisors to be nilpotent or idempotent are also established.

Abstract

An element in a ring $R$ is called uniquely weakly nil-clean if every element in $R$ can be uniquely written as a sum or a difference of a nilpotent and an idempotent in the sense of very idempotents. The structure of the ring in which every zero-divisor is uniquely weakly nil-clean is completely determined. We prove that every zero-divisor in a ring $R$ is uniquely weakly nil-clean if and only if $R$ is a D-ring, or $R$ is abelian, periodic, and $R/J(R)$ is isomorphic to a field $F$, ${\Bbb Z}_{3}\oplus {\Bbb Z}_{3}$, ${\Bbb Z}_{3}\oplus B$ where $B$ is Boolean, or a Boolean ring. As a specific case, rings in which every zero-divisor $a$ or $-a$ is a nilpotent or an idempotent are also considered. Furthermore, we prove that every zero-divisor in a ring $R$ is uniquely nil-clean if and only if $R$ is a D-ring, or $R$ is abelian, periodic; and $R/J(R)$ is Boolean.\vskip3mm \no {\bf Key words}: Zero-divisor; Uniquely weakly nil-clean ring; Uniquely nil-clean ring.