Extensions of Regular Rings
محورهای موضوعی : مجله بین المللی ریاضیات صنعتیSH. A. Safari ‎Sabet‎ 1 , M. Farmani 2
1 - Department of Mathematics, Central Tehran Branch, Islamic Azad University, Tehran, Iran
2 - Young Researchers and Elite Club, Roudehen Branch, Islamic Azad University, Roudehen, Iran
کلید واژه: Group ring, $pi$-Regular, $mathbb{Z}G$-Regular, Strongly $mathbb{Z}G$-regular,
چکیده مقاله :
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.
