Extensions of Regular ‎Rings‎

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+۱)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.‎

Keywords:

Group ring , pi-Regular , mathbb{Z}G-Regular , Strongly mathbb{Z}G-‎regular