Some properties of $e$-symmetric rings

Authors : F M, Junchao WEI

Pages : 2389-2399

View : 12 | Download : 7

Publication Date : 0000-00-00

Article Type : Research Paper

Abstract :In this paper, we first give some characterizations of $e$-symmetric rings. We prove that $R$ is an $e$-symmetric ring if and only if $a_{1}a_{2}a_{3}=0$ implies that $a_{\sigmainsert ignore into journalissuearticles values(1);}a_{\sigmainsert ignore into journalissuearticles values(2);}a_{\sigmainsert ignore into journalissuearticles values(3);}e=0$, where $\sigma $ is any transformation of $\{1,2,3\}$. With the help of the Bott--Duffin inverse, we show that for $e\in ME_{l}insert ignore into journalissuearticles values(R);$, $R$ is an $e$-symmetric ring if and only if for any $a\in R$ and $g\in Einsert ignore into journalissuearticles values(R);$, if $a$ has a Bott--Duffin $insert ignore into journalissuearticles values(e,g);$-inverse, then $g=eg$. Using the solution of the equation $axe=c$, we show that for $e\in ME_{l}insert ignore into journalissuearticles values(R);$, $R$ is an $e$-symmetric ring if and only if for any $a,c \in R$, if the equation $axe=c$ has a solution, then $c=ec$. Next, we study the properties of $e$-symmetric $*$-rings. Finally we discuss when the upper triangular matrix ring $T_{2}insert ignore into journalissuearticles values(R);$ insert ignore into journalissuearticles values(resp. $T_{3}insert ignore into journalissuearticles values(R,I);$); becomes an $e$-symmetric ring, where $e\in Einsert ignore into journalissuearticles values(T_{2}insert ignore into journalissuearticles values(R););$ insert ignore into journalissuearticles values(resp. $e\in Einsert ignore into journalissuearticles values(T_{3}insert ignore into journalissuearticles values(R,I););$);.
Keywords : e Symmetric ring, ring, left semicentral, left min abel ring, Bott Duffin inverse, upper triangular matrix ring