Strongly Rickart Rings

Abstract

Let R be a ring with identity. In this paper we introduce a strongly Rickart ring as a stronger concept of a Rickart ring. A ring R is said to be strongly Rickart ring if the right annihilators of each single element in R is generated by a left semicentral idempotent in R. This class of rings is proper class in right Rickart rings, p.q.-Baer rings, reduced rings and semiprime rings. The relation between strongly Rickart and strongly regular are studied. We discuss some types of extension of strongly Rickart ring such as the Dorroh extension and the idealization ring.

Key words: strongly Rickart ring, Rickart ring, right annihilator element, semicentral idempotent element, Dorroh extension, idealization of a module.