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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 SF-rings are generalized.

On a common generalization of symmetric rings and quasi duo rings