# On Zhou nil-clean rings

## TL;DR

This paper characterizes Zhou nil-clean rings, a class where elements are sums of tripotents and nilpotents, by exploring their relations with polynomials, idempotents, and 2-idempotents, and establishing conditions for their structure.

## Contribution

It provides new characterizations of Zhou nil-clean rings through polynomial relations and idempotent decompositions, linking them to Kosan exchange rings.

## Key findings

- Existence of polynomial-based idempotent decompositions in Zhou nil-clean rings.
- Characterization of these rings via sums of 2-idempotents and nilpotents.
- Equivalence of certain polynomial conditions with the ring being a Kosan exchange ring.

## Abstract

A ring R is a Zhou nil-clean ring if every element in R is the sum of two tripotents and a nilpotent that commute. In this paper, Zhou nil-clean rings are further discussed with an emphasis on their relations with polynomials, idempotents and 2- idempotents. any a \in R, there exists e 2 Z[a] such that a-e \in R is nilpotent and e^5 = 5e^3 -4e, if and only if for any a 2 R, there exist idempotents e; f; g; h 2 Z[a] and a nilpotent w such that a = e+f +g+h+w, if and only if for any a 2 R, there exist 2-idempotents e; f 2 Z[a] and a nilpotent w 2 R such that a = e + f + w, if and only if for any a 2 R, there exists a 2-idempotent e 2 Z[a] and a nilpotent w 2 R such that a^2 = e+w, if and only if R is a Kosan exchange ring.