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...

Let $R$ be an associative ring and $M$ be a left $R$-module. We introduce the concept of the incidence module $I(X,M)$ of a locally finite partially ordered set $X$ over $M$. We study the properties of $I(X,M)$ and give the necessary and sufficient conditions for the incidence module to be an IN-module, -module, nil injective module and nonsingular module, respectively. Furthermore, we show that the class of -modules is closed under direct product and upper triangular matrix modules.

Si considerano le estensioni chiuse $B$ di un $R$-modulo $A$ mediante un $R$-modulo $C$ nel caso in cui $R$ sia un anello semi-artiniano, cioè un anello $R$ con la proprietà che per ogni quoziente $(R{/}_I)\ne 0$ sia soc $(R/I)\ne 0$. Tali estensioni sono caratterizzate dal fatto che $A$ deve essere un sottomodulo semi-puro di $B$.

We provide some characterizations of rings $R$ for which every (finitely generated) module belonging to a class $𝒞$ of $R$-modules is a direct sum of cyclic submodules. We focus on the cases, where the class $𝒞$ is one of the following classes of modules: semiartinian modules, semi-V-modules, V-modules, coperfect modules and locally supplemented modules.