On quasi-Frobenius bimodules and corings

Frobenius bimodules are connected with Frobenius algebras and extensions. For instance, a ring extension φ : R → S is a Frobenius extension if and only if RSS is a Frobenius bimodule [1]. Brzeziński and Gómez-Torrecillas studied in [4] certain properties of comatrix corings in relation to properties of bimodules. In particular they showed that the comatrix coring [6] induced by any Frobenius bimodule is a Frobenius coring. Recently, Castaño Iglesias and Năstăsescu [5] studied quasi-Frobenius extensions from the viewpoint of quasi-Frobenius functors. The ring extension φ : R → S is quasi-Frobenius (left and right extension) if and only if the restriction of scalars functor φ∗ : SM → RM is a quasi-Frobenius functor (Remark 1.2). Following this direction, we shall extend some results in [4] to the case when RMS is a quasi-Frobenius bimodule. Using the notion of quasi-Frobenius functor and properties on quasi-Frobenius corings given in [5], we discuss the questions of when a quasi-Frobenius bimodule induce a quasi-Frobenius extension and a quasi-Frobenius coring.