ON REVERSIBILITY RELATED TO IDEMPOTENTS

ON REVERSIBILITY RELATED TO IDEMPOTENTS

This article concerns a ring property which preserves the reversibility of elements at nonzero idempotents. A ring R shall be said to be quasi-reversible if $0{\neq}ab{\in}I(R)$ for a, $b{\in}R$ implies $ba{\in}I(R)$, where I(R) is the set of all idempotents in R. We investigate the quasi-reversibility of 2 by 2 full and upper triangular matrix rings over various kinds of reversible rings, concluding that the quasi-reversibility is a proper generalization of the reversibility. It is shown that the quasi-reversibility does not pass to polynomial rings. The structure of Abelian rings is also observed in relation with reversibility and quasi-reversibility.