Strongly Nil-*-Clean Rings

Abstract

A *-ring $R$ is called a strongly nil-*-clean ring if every element of $R$ is the sum of a projection and a nilpotent element that commute with each other. In this article, we show that $R$ is a strongly nil-*-clean ring if and only if every idempotent in $R$ is a projection, $R$ is periodic, and $R/J(R)$ is Boolean. For any commutative *-ring $R$, we prove that the algebraic extension $R[i]$ where $i^2=\mu i+\eta$ for some $\mu,\eta\in R$ is strongly nil-*-clean if and only if $R$ is strongly nil-*-clean and $\mu\eta$ is nilpotent. The relationships between Boolean *-rings and strongly nil-*-clean rings are also obtained.