A well known result of B. Osofsky assertsthat if R is a left (or right) perfect, left and right selfinjective ring thenR is quasi-Frobenius. It was subsequently conjectured by Carl Faith that every left(or right) perfect, left selfinjective ring is quasi-Frobenius. While several authors have proved theconjecture in the affirmative under some restricted chain conditions, the conjecture remains open evenif R is a semiprimary, local, left selfinjective ring withJ(R)^3=0. In this paper we construct a local ring R withJ(R)^3=0 and characterize when R is artinian or selfinjective interms of conditions on a bilinear mapping from a D-D-bimodule toD, where D is isomorphic to R/J(R). Our workshows that finding a counterexample to the Faith conjecture depends on the existence of aD-D-bimodule over a division ring Dsatisfying certain topological conditions.