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Turkish Journal of Mathematics

F M
JUNCHAO WEI

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 $a_{\sigma(1)}a_{\sigma(2)}a_{\sigma(3)}e=0$, where $\sigma $ is any transformation of $\{1,2,3\}$. With the help of the Bott--Duffin inverse, we show that for $e\in ME_{l}(R)$, $R$ is an $e$-symmetric ring if and only if for any $a\in R$ and $g\in E(R)$, if $a$ has a Bott--Duffin $(e,g)$-inverse, then $g=eg$. Using the solution of the equation $axe=c$, we show that for $e\in ME_{l}(R)$, $R$ 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 $T_{2}(R)$ (resp. $T_{3}(R,I)$) becomes an $e$-symmetric ring, where $e\in E(T_{2}(R))$ (resp. $e\in E(T_{3}(R,I))$).

DOI

10.3906/mat-1709-10

Keywords

$e$-Symmetric ring, $*$-ring, left semicentral, left min-abel ring, Bott--Duffin inverse, upper triangular matrix ring

First Page

2389

Last Page

2399

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