NJ-semicommutative Rings

. We call a ring R NJ-semicommutative if wh ∈ N ( R ) implies wRh ⊆ J ( R ) for any w , h ∈ R . The class of NJ-semicommutative rings is large enough that it contains semicom-mutative rings, left (right) quasi-duo rings, J-clean rings, and J-quasipolar rings. We provide some conditions for NJ-semicommutative rings to be reduced. We also observe that if R / J ( R ) is reduced, then R is NJ-semicommutative, and therefore we provide some conditions for NJ-semicommutative ring R for which R / J ( R ) is reduced. We also study some extensions of NJ-semicommutative rings wherein, among other results, we prove that the polynomial ring over an NJ-semicommutative ring need not be NJ-semicommutative.