Equivalences induced by adjoint functors

Let A and B be two Grothendieck categories, R : A --> B, L: B --> A a pair of adjoint functors, S is an element of B a generator, and U= L(S). U defines a hereditary torsion class in A, which is carried by L, under suitable hypotheses, into a hereditary torsion class in B. We investigate necessary and sufficient conditions which assure that the functors R and L induce equivalences between the quotient categories of A and B modulo these torsion classes. Applications to generalized module categories, rings with local units and group graded rings are also given here.