Revisiting J-semicommutative rings

Let J(R) denote the Jacobson radical of a ring R. R is called J-semicommutative if for any \(a,b\in R, ab=0\) $a,b\in R,ab=0$ implies \(aRb\subseteq J(R)\) $aRb\subseteq J(R)$ . We observe that the class of J -semicommutative rings contains the class of left (right) quasi-duo rings and various existing versions of semicommutative rings, symmetric rings and reversible rings. We provide some conditions for J-semicommutative rings to be left quasi-duo. Finally, the consequences of J-semicommutativity conditions over some classes of rings are discussed.