### $*$-biregular rings

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Let G(X) denote the smallest (von Neumann) regular ring of real-valued functions with domain X that contains C(X), the ring of continuous real-valued functions on a Tikhonov topological space (X,τ). We investigate when G(X) coincides with the ring $C(X,{\tau}_{\delta})$ of continuous real-valued functions on the space $(X,{\tau}_{\delta})$, where ${\tau}_{\delta}$ is the smallest Tikhonov topology on X for which $\tau \subseteq {\tau}_{\delta}$ and $C(X,{\tau}_{\delta})$ is von Neumann regular. The compact and metric spaces for which $G\left(X\right)=C(X,{\tau}_{\delta})$ are characterized. Necessary, and different sufficient, conditions...

We show in an additive inverse regular semiring $(S,+,\xb7)$ with ${E}^{\u2022}\left(S\right)$ as the set of all multiplicative idempotents and ${E}^{+}\left(S\right)$ as the set of all additive idempotents, the following conditions are equivalent: (i) For all $e,f\in {E}^{\u2022}\left(S\right)$, $ef\in {E}^{+}\left(S\right)$ implies $fe\in {E}^{+}\left(S\right)$. (ii) $(S,\xb7)$ is orthodox. (iii) $(S,\xb7)$ is a semilattice of groups. This result generalizes the corresponding result of regular ring.

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...

This article presents a brief survey of the work done on rings generated by their units.

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...