  In this paper, we investigate some properties of annihilator $(b,c)$-inverses
in an arbitrary ring. We demonstrate that one-sided annihilator
$(b,c)$-inverses of elements in arbitrary rings may behave differently in
contrast to one-sided $(b,c)$-inverses. Also, we discuss intertwining property,
absorption law, reverse order law, and Cline's formula for annihilator
$(b,c)$-inverses. As applications, we improve and extend some known results to
$(b,c)$-inverses. In particular, we derive an equivalent condition of
intertwining property for $(b,c)$-inverses in semigroups.
