An application of the Zhou radical to the \(e\)-reversibility of rings

Abstract

Let \(R\) be a ring and \(e\) be an idempotent element of \(R\). The Zhou radical of \(R\) denoted by \(\delta(R)\) is the intersection of maximal essential right ideals of \(R\). In the literature, \(e\)-reversible rings were studied regarding the question of how idempotent elements affect the reversible property of rings. In this paper, we provide an application of the Zhou radical of a ring to the reversibility depending on idempotents. Accordingly, we study rings \(R\) in which \(ab = 0\) implies \(bae\in\delta(R)\) (or \(eba\in\delta(R),\)) where \(a, b\in R\), called Zhou right (or left) \(e\)-reversible rings. Besides studying the structure of Zhou \(e\)-reversible rings, we investigate relations between Zhou \(e\)-reversible rings and some known rings, such as Zhou \(e\)-reduced rings, central reversible rings, NI-rings, semiperfect rings and matrix rings. In addition to these, we determine the Zhou radical of some certain rings, such as Morita context rings and special subrings of a direct product of rings.