Extensions of Regular ‎Rings‎

Subject Areas : International Journal of Industrial Mathematics

SH. 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

Received: 2016-11-06 Accepted : 2016-11-06 Published : 2016-11-01

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

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.‎