# On graded nil clean rings

## TL;DR

This paper introduces and investigates graded nil clean rings, exploring their properties, extensions to matrix and group rings, and conditions for nil cleanness to be preserved in graded structures.

## Contribution

It defines graded nil clean rings, studies their extensions, and examines conditions for nil cleanness to imply graded nil cleanness in various graded ring contexts.

## Key findings

- Characterization of graded nil clean rings
- Conditions for nil cleanness to imply graded nil cleanness
- Extensions to graded matrix and group rings

## Abstract

In this paper we introduce and study the notion of a graded (strongly) nil clean ring which is group graded. We also deal with extensions of graded (strongly) nil clean rings to graded matrix rings and to graded group rings. The question of when nil cleanness of the component, which corresponds to the neutral element of a group, implies graded nil cleanness of the whole graded ring is examined. Similar question is discussed in the case of groupoid graded rings as well.