Rings whose elements are sums of m-potents and nilpotents

Abstract

For a fixed natural number an element a of a ring R is m-nil clean if where 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 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.