Rings in Which Element is a Sum of the Transformation Elements of Idempotents and Special Elements

In this article, further generalizations are made for the nil clean ring and the ur‐clean ring, obtained as extensions of the clean ring. Firstly, consider rings where each element can be expressed as n idempotents plus one nilpotent, any two commute under multiplication. Some properties of this ring and its relationship to some special rings will be discussed in this paper. Secondly, replace the idempotent element that make up the ur‐clean ring with the quasi‐idempotent element to obtain ur‐quasi‐clean ring. Finally, the ur‐quasi‐cleanness will be considered on the involution rings, define by ∗ ur‐quasi‐clean rings.