Narrow Results

A ring $R$ is strongly 2-nil-clean if every element in $R$ is the sum of two idempotents 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 $a\in R$, $a-a^3\in R$ is nilpotent, if and only if for all $a\in R$, $a^2\in R$ is strongly nil-clean, if and only if every element in $R$ is the sum of a tripotent and a nilpotent that commute. Furthermore, we prove that a ring $R$ is strongly 2-nil-clean if and only if $R/J(R)$ is tripotent and $J(R)$ is nil, if and only if $R\cong R_1,R_2$ 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. Strongly 2-nil-clean group algebras are investigated as well.

We completely determine the rings for which every element is a sum of a nilpotent, an idempotent and a tripotent that commute with one another, and the rings for which every element is a sum of a nilpotent and two tripotents that commute with one another.

A ring $R$ is strongly 2-nil-clean if every element in $R$ is the sum of a tripotent and a nilpotent that commute. We prove that a ring $R$ is strongly 2-nil-clean if and only if $R$ is a strongly feebly clean 2-UU ring if and only if $R$ is an exchange 2-UU ring. Furthermore, we characterize strongly 2-nil-clean ring via involutions. We show that a ring $R$ is strongly 2-nil-clean if and only if every element in $R$ is the sum of an idempotent, an involution and a nilpotent that commute.

A ring $R$ is Zhou nil-clean if every element in $R$ is the sum of two tripotents and a nilpotent that commute. A ring $R$ is feebly clean if for any $a\in R$ there exist two orthogonal idempotents $e,f\in R$ and a unit $u\in R$ such that $a=e-f+u$. In this paper, Zhou nil-clean rings are further discussed with an emphasis on their relations with feebly clean rings. We prove that a ring $R$ is Zhou nil-clean if and only if $R$ is feebly clean, $J(R)$ is nil and $U(R/J(R))$ has exponent $\le 4$ if and only if $R$ is weakly exchange, $J(R)$ is nil and $U(R/J(R))$ has exponent $\le 4$. New properties of Zhou rings are thereby obtained.