Rings with annihilator chain conditions and right distributive rings

Abstract

We prove that if a right distributive ring R R , which has at least one completely prime ideal contained in the Jacobson radical, satisfies either a.c.c or d.c.c. on principal right annihilators, then the prime radical of R R is the right singular ideal of R R and is completely prime and nilpotent. These results generalize a theorem by Posner for right chain rings.