  Let $R$ be a ring and $b, c\in R$. In this paper, we give some
characterizations of the $(b,c)$-inverse, in terms of the direct sum
decomposition, the annihilator and the invertible elements. Moreover, elements
with equal $(b,c)$-idempotents related to their $(b, c)$-inverses are
characterized, and the reverse order rule for the $(b,c)$-inverse is
considered.

Rings and Algebras