  In this paper we introduce a new generalized inverse in a ring -- one-sided
$(b, c)$-inverse, derived as an extension of $(b, c)$-inverse. This inverse
also generalizes one-sided inverse along an element, which was recently
introduced by H. H. Zhu et al. [H. H. Zhu, J. L. Chen, P. Patr\'{i}cio, Further
results on the inverse along an element in semigroups and rings, Linear
Multilinear Algebra, 64 (3) (2016) 393-403]. Also, here we present one-sided
annihilator $(b, c)$-inverse, which is an extension of the annihilator $(b,
c)$-inverse. Necessary and sufficient conditions for the existence of these new
generalized inverses are obtained. Furthermore, we investigate conditions for
the existence of one-sided $(b, c)$-inverse of a product of three elements and
we consider some properties of one-sided $(b, c)$-inverses.
