Characterizations and representations of left and right hybrid ( b , c )-inverses in rings

Let R be an associative ring with unity 1 and let $$a,b,c\in R$$ . In this paper, several characterizations for left and right hybrid (b, c)-inverses of a are derived. Moreover, their formulae are given by regularities of certain elements. Then, we give characterizations of right (b, c)-inverses and right annihilator (b, c)-inverses of the product of three elements. Finally, relations among the right hybrid (b, c)-inverse of paq, the right (qb, c)-inverse of aq and the right annihilator (b, cp)-inverse of aq are given.