An Equivalent Condition and Some Properties of Strong J-Symmetric Ring

Abstract

Let $J\left(R\right)$ denote the Jacobson radical of a ring $R$ . We say that ring $R$ is strong J-symmetric if, for any $a,b,c\in R$ , $abc\in J\left(R\right)$ implies $bac\in J\left(R\right)$ . If ring $R$ is strong J-symmetric, then it is proved that $R\left[x\right]/\left(x^n\right)$ is strong J-symmetric for any $n\ge 2$ . If $R$ and $S$ are rings and ${}_R{{\displaystyle W}}_S$ is a $\left(R,S\right)$ -bimodule, $E=T\left(R,S,W\right)=\left(\begin{array}{cc}R& W\\ 0& S\end{array}\right)=\left\{\left(\begin{array}{cc}r& w\\ 0& s\end{array}\right)|r\in R,w\in W,s\in S\right\},$ then it is proved that $R$ and $S$ are J-symmetric if and only if $E$ is J-symmetric. It is also proved that $R$ and $S$ are strong J-symmetric if and only if $E$ is strong J-symmetric.