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.‎ تفاصيل المقالة