A ring R is called GWCN if x² y² = xy² x for all x ∈ N(R) and y ∈ R, which is a proper generalization of reduced rings and CN rings. We study the sufficient conditions for GWCN rings to be reduced and CN. We first discuss many properties of GWCN rings. Next, we give some interesting characterizations of left min-abel rings. Finally, with the help of exchange GWCN rings, we obtain some characterizations of strongly regular rings.