On a common generalization of symmetric rings and quasi duo rings

Let $J(R)$ denote the Jacobson radical of a ring $R$. We call a ring $R$ as $J$-symmetric if for any $a,b, c\in R$, $abc=0$ implies $bac\in J(R)$. It turns out that $J$-symmetric rings are a common generalization of left (right) quasi-duo rings and generalized weakly symmetric rings. Various properties of these rings are established and some results on exchange rings and the regularity of left $\mathrm{SF}$-rings are generalized.