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Keywords:
nilpotent; projection; $*$-tripotent ring; symmetry; strongly $*$-clean ring
nilpotent; projection; $*$-tripotent ring; symmetry; strongly $*$-clean ring
Summary:
A $*$-ring $R$ is strongly 2-nil-$*$-clean if every element in $R$ is the sum of two projections and a nilpotent that commute. Fundamental properties of such $*$-rings are obtained. We prove that a $*$-ring $R$ is strongly 2-nil-$*$-clean if and only if for all $a\in R$, $a^2\in R$ is strongly nil-$*$-clean, if and only if for any $a\in R$ there exists a $*$-tripotent $e\in R$ such that $a-e\in R$ is nilpotent and $ea=ae$, if and only if $R$ is a strongly $*$-clean SN ring, if and only if $R$ is abelian, $J(R)$ is nil and $R/J(R)$ is $*$-tripotent. Furthermore, we explore the structure of such rings and prove that a $*$-ring $R$ is strongly 2-nil-$*$-clean if and only if $R$ is abelian and $R\cong R_1, R_2$ or $R_1\times R_2$, where $R_1/J(R_1)$ is a $*$-Boolean ring and $J(R_1)$ is nil, $R_2/J(R_2)$ is a $*$-Yaqub ring and $J(R_2)$ is nil. The uniqueness of projections of such rings are thereby investigated.
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