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We prove that if a right distributive ring 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 is the right singular ideal of R and is completely prime and nilpotent. These results generalize a theorem by Posner for right chain rings.

Rings with annihilator chain conditions and right distributive rings