Rings whose elements are sums of m-potents and nilpotents

Abstract

For a fixed natural number $m\ge 2,$ an element a of a ring R is m-nil clean if $a=e+b$ where $e^m=e\in R$ and b is nilpotent; if further eb = be, a is called strongly m-nil clean. The ring R is called m-nil clean (resp., strongly m-nil clean) if each of its elements is m-nil clean (resp., strongly m-nil clean). For a fixed natural number $m\ge 2,$ the strongly m-nil clean rings are a big subclass of the periodic rings which contains the class of strongly nil clean rings. The strongly m-nil cleanness of the tensor product of algebras, matrix rings, Morita contexts, and group rings is discussed in detail. Our results extend several existing results. Examples are provided to illustrate our results.

Keywords:

• m-nil clean rings
• strongly m-nil clean rings
• nil clean rings
• clean rings

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 16U99
• 16Z05
• 16S34

Acknowledgement

We would like to thank the referee for their carefully reading the paper and his/her helpful suggestions which improved its quality, e.g., Theorem 1.4. The work of the third author was supported under the development program of the Volga Region Mathematical Center (agreement no.075-02-2020-1478).