# Formal triangular matrix ring with nil clean index $4$

## TL;DR

This paper characterizes formal triangular matrix rings with a nil clean index of 4, providing insights into their algebraic structure and properties related to idempotents and nilpotent elements.

## Contribution

It offers a complete characterization of formal triangular matrix rings with nil clean index 4, advancing understanding of their algebraic structure.

## Key findings

- Characterization of formal triangular matrix rings with nil clean index 4
- Analysis of idempotent elements in these rings
- Insights into the structure of nilpotent elements in the rings

## Abstract

For an element $a \in R$, let $\eta(a)=\{e\in R\mid e^2=e\mbox{ and }a-e\in \mbox{nil}(R)\}$. The nil clean index of $R$, denoted by NinA$(R)$, is defined as Nin$(R)=\sup \{\mid \eta(a)\mid: a\in R\}$. In this article we have characterized formal triangular matrix ring $\begin{pmatrix}A & M0 & B\end{pmatrix}$ with nil clean index $4$.