Distributive semiprime rings

It is proved that a right distributive semiprime \textrm{PI} ring $A$ is a left distributive ring and for each element $x\in A$ there is a positive integer $n$ such that $x^nA=Ax^n$. We describe both right distributive right Noetherian rings algebraic over the center of the ring and right distributive left Noetherian \textrm{PI} rings. We also characterize rings all of whose Pierce stalks are right chain right Artin rings.