MMN-2000
Some notes on first strongly graded rings
Rashid Abu-Dawwas; Ala'a Mesleh; Khaldoun Al-Zoubi;Abstract
Let $G$ be a group with identity $e$ and $R$ be an associative ring with a nonzero unity $1$. Assume that $R$ is first strongly $G$-graded and $H=supp(R, G)$. For $g\in H$, define $\alpha_{g}(x)=\displaystyle\sum_{i=1}^{n_{g}}r_{g}^{(i)}xt^{(i)}_{g^{-1}}$ where $x\in C_{R}(R_{e})=\left\{r\in R:rx=xr\mbox{ for all }x\in R_{e}\right\}$, $r_{g}^{(i)}\in R_{g}$ and $t_{g^{-1}}^{(i)}\in R_{g^{-1}}$ for all $i=1,....., n_{g}$ for some positive integer $n_{g}$. In this article, we study $\alpha_{g}(x)$ and it's properties.
Vol. 18 (2017), No. 2, pp. 547-554