Abstract
For an associative ring R with identity, we define an element in R to be NR-clean if it is the sum of a (von Neumann) regular and a nilpotent. R is called an NR-clean ring if every element of R is NR-clean. This class of rings lies properly between nil clean rings and UR rings. In this paper, we give many examples and properties of NR-clean rings and investigate the behavior of these properties under various ring extensions. We also define and study uniquely NR-clean rings and give a characterization of such rings. Finally, we justify some conditions under which the group rings are NR-clean.
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