On some extensions of strongly unit nil-clean rings

Abstract

An element $x\in R$ is considered (strongly) nil-clean if it can be expressed as the sum of an idempotent $e\in R$ and a nilpotent $b\in R$ (where $eb=be$). If for any $x\in R$, there exists a unit $u\in R$ such that $ux$ is (strongly) nil-clean, then $R$ is called a (strongly) unit nil-clean ring. It is worth noting that any unit-regular ring is strongly unit nil-clean. In this note, we provide a characterization of the unit regularity of a group ring, along with an additional condition. We also fully characterize the unit-regularity of the group ring ${\mathbb{Z}}_{n}G$ for every $n>1$. Additionally, we discuss strongly unit nil-cleanness in the context of Morita contexts, matrix rings, and group rings.

12 Pages, to appear in