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 $\mathbb{Z}_nG$ for every $n > 1$. Additionally, we discuss strongly unit nil-cleanness in the context of Morita contexts, matrix rings, and group rings.

Theorems & Definitions (59)

• Theorem 1.1: Connell
• Theorem 1.2: Woods
• Definition 2.1
• Lemma 2.2
• proof
• Theorem 2.3
• proof
• Proposition 3.1
• proof
• Lemma 3.2