Quasi-Frobenius corings and quasi-Frobenius extensions

In this article, the notions of a Frobenius pair of functors and Frobenius corings are generalized to an l -QF pair of functors and l -QF corings. We prove that an extension iota: B -> A is left quasi-Frobenius if and only if (F-1 , G(1)) is an l-QF pair of functors, where F-1 : M-A -> M-B is the restriction of scalars functors, and G(1) = A circle times(B-) : M-B -> M-A is the induction functor. For an A-coring C, we prove that C is an l -QF coring if and only if A -> *C is an l -QF extension and A is a finitely generated projective modules if and only if (G(2) , F-2) is an l -QF pair of functors, where G(2) = C circle times(A-) : M-A -> M-C is the induction functor, F-2 : M-C -> M-A is the forgetful functor, the result of Brzezinski is generalized.