Extensions of Regular ‎Rings‎

Abstract

Let $R$ be an associative ring with identity. An element $x in R$ is called $mathbb{Z}G$-regular (resp. strongly $mathbb{Z}G$-regular) if there exist $g in G$, $n in mathbb{Z}$ and $r in R$ such that $x^{ng}=x^{ng}rx^{ng}$ (resp. $x^{ng}=x^{(n+1)g}$). A ring $R$ is called $mathbb{Z}G$-regular (resp. strongly $mathbb{Z}G$-regular) if every element of $R$ is $mathbb{Z}G$-regular (resp. strongly $mathbb{Z}G$-regular). In this paper, we characterize $mathbb{Z}G$-regular (resp. strongly $mathbb{Z}G$-regular) rings. Furthermore, this paper includes a brief discussion of $mathbb{Z}G$-regularity in group ‎rings.‎