Plane modules and distributive rings

Abstract

Let $A$ be a semiprime ring entire over its center. We prove that the following conditions are equivalent:
(a) A is a ring distributive from the right (left);
(b) w.gl. $\dim (A) ≤ 1$; moreover, if $M$ is an arbitrary prime ideal of the ring $A$, then $A/M$ is a right Ore set.