In this paper, we extend the notion of annihilator (b, c)-inverses to arbitrary rings (not necessary with identity). Then we demonstrate that annihilator (b, c)-inverses of elements in arbitrary rings may behave differently in contrast to (b, c)-inverses in semigroups. Further connections between annihilator (b, c)-inverses and one-sided ones are also investigated. In addition, we obtain a new general case of bicommutant property for annihilator (b, c)-inverses. As a consequence, we show that  and  together imply  where  is the Moore-Penrose inverse of a in a *-ring.

generalized inverse, one-sided inverse, bicommutant.