Some Classes of Weakly Prime Center Rings

Let R be a ring. The ring R is called weakly prime center(WPC ring) if ab is an element of Z(R) implies that aRb is an ideal of R. In this paper, we prove that every left(right) duo ring is a WPC ring. Also we introduce some classes of rings with nilpotent Jacobson radical which are WPC rings. Finally, we prove that a simple ring is a WPC ring if and only if it is a domain.