Generalized semicommutative rings

We call a ring R generalized semicommutative if for any $a, b \in R, ab = 0$ implies there exists positive integers $m, n$ such that $a^mRb^n = 0$. We observe that the class of generalized semicommutative rings strictly lies between the class of central semicommutative rings and weakly semicommutative-I rings. Relationships are provided between generalized semicommutative rings and some known classes of rings. From an arbitrary generalized semicommutative ring, we produce many families of generalized semicommutative rings. Finally we provide some conditions for a generalized semicommutative ring to be reduced.