Characterizations of the $(b,c)$-inverse in a ring

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.