Symmetric property of rings with respect to the Jacobson radical
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Let $R$ be a ring with identity and $J(R)$ denote the Jacobsonradical of $R$, i.e., the intersection of all maximal left ideals of $R$. A ring $R$ is called {\it $J$-symmetric} if for any $a,b,c\in R$, $abc = 0$ implies $bac \in J(R)$. We prove that some results of symmetric rings can be extended to the $J$-symmetric rings for this general setting. We give many characterizations of such rings. We show that the class of $J$-symmetric rings lies strictly between the class of symmetric rings and the class of directly finite rings.
Abstract
Let $R$ be a ring with identity and $J(R)$ denote the Jacobsonradical of $R$, i.e., the intersection of all maximal left ideals of $R$. A ring $R$ is called {\it $J$-symmetric} if for any $a,b,c\in R$, $abc = 0$ implies $bac \in J(R)$. We prove that some results of symmetric rings can be extended to the $J$-symmetric rings for this general setting. We give many characterizations of such rings. We show that the class of $J$-symmetric rings lies strictly between the class of symmetric rings and the class of directly finite rings.
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