E-reversibility of rings via quasinilpotents

Abstract

Abstract The reversible property of rings was introduced by Cohn and has important generalizations in noncommutative ring theory. In this paper, reversibility of rings is investigated in relation with quasinilpotents and idempotents, and our argument is spread out based on this. We call a ring R Qnil e-reversible if for any a , b ∈ R {a,b\in R} , being a ⁢ b = 0 {ab=0} implies b ⁢ a ⁢ e ∈ R qnil {bae\in R^{\rm qnil}} for a prescribed idempotent e ∈ R {e\in R} , where R qnil {R^{\rm qnil}} denotes the set of all quasinilpotent elements of R. In the first, we determine the set R qnil {R^{\rm qnil}} for some classes of rings to investigate the structure of Qnil e-reversible rings. In the second, we use R qnil {R^{\rm qnil}} to define Qnil e-reversibility of rings. The notion of Qnil e-reversible ring is a proper generalization of that of e-semicommutative ring, Qnil-semicommutative ring, e-reversible ring and right (left) quasi-duo ring. We obtain some relations between a ring and its quotient rings in terms of Qnil e-reversibility. Applications via some ring extensions and examples illustrating our results are provided.