We introduce a class of rings which is a generalization of reflexive rings and $J$-reversible rings. Let $R$ be a ring with identity and $J\left(R\right)$ denote the Jacobson radical of $R$. A ring $R$ is called $J$-reflexive if for any $a,b\in R$, $aRb=0$ implies $bRa\subseteq J\left(R\right)$. We give some characterizations of a $J$-reflexive ring. We prove that some results of reflexive rings can be extended to $J$-reflexive rings for this general setting. We conclude some relations between $J$-reflexive rings and some related rings. We investigate some extensions of...

In this paper, we introduce a subclass of strongly clean rings. Let $R$ be a ring with identity, $J$ be the Jacobson radical of $R$, and let $J^{\#}$ denote the set of all elements of $R$ which are nilpotent in $R/J$. An element $a\in R$ is called very $J^{\#}$-clean provided that there exists an idempotent $e\in R$ such that $ae=ea$ and $a-e$ or $a+e$ is an element of $J^{\#}$. A ring $R$ is said to be very $J^{\#}$-clean in case every element in $R$ is very $J^{\#}$-clean. We prove that every very $J^{\#}$-clean ring is strongly $\pi$-rad clean and has stable range one. It is shown...

A ring $R$ is defined to be left almost Abelian if $ae=0$ implies $aRe=0$ for $a\in N\left(R\right)$ and $e\in E\left(R\right)$, where $E\left(R\right)$ and $N\left(R\right)$ stand respectively for the set of idempotents and the set of nilpotents of $R$. Some characterizations and properties of such rings are included. It follows that if $R$ is a left almost Abelian ring, then $R$ is $\pi$-regular if and only if $N\left(R\right)$ is an ideal of $R$ and $R/N\left(R\right)$ is regular. Moreover it is proved that (1) $R$ is an Abelian ring if and only if $R$ is a left almost Abelian left idempotent reflexive ring. (2) $R$ is strongly...