# Matrices over Zhou nil-clean rings

## TL;DR

This paper investigates matrices over Zhou nil-clean rings, proving that if such rings are 2-primal with bounded index, then every square matrix can be expressed as the sum of two tripotents and a nilpotent, expanding the class of rings with this property.

## Contribution

It establishes that matrices over 2-primal Zhou nil-clean rings of bounded index can be decomposed into tripotents and nilpotents, broadening understanding of matrix decompositions in ring theory.

## Key findings

- Matrices over certain Zhou nil-clean rings can be decomposed into tripotents and nilpotents.
- The class of rings where such decompositions are possible is expanded.
- Provides structural insights into matrices over Zhou nil-clean rings.

## Abstract

A ring R is Zhou nil-clean if every element in R is the sum of two tripotents and a nilpotent that commute. Let R be a Zhou nil-clean ring. If R is 2-primal (of bounded index), we prove that every square matrix over R is the sum of two tripotents and a nilpotent. These provides a large class of rings over which every square matrix has such decompositions by tripotent and nilpotent matrices.