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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.

CHARACTERIZATIONS OF ANNIHILATOR (b, c)-INVERSES IN ARBITRARY RINGS