A look at the faith conjecture

A well known result of B. Osofsky asserts that if R is a left (or right) perfect, left and right selfinjective ring then R 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 the conjecture in the affirmative under some restricted chain conditions, the conjecture remains open even if R is a semiprimary, local, left selfinjective ring with J(R)3 = 0. In this paper we construct a local ring R with J(R)3 = 0 and characterize when R is artinian or selfinjective in terms of conditions on a bilinear mapping from a D-D-bimodule to D, where D is isomorphic to R/J(R). Our work shows that finding a counterexample to the Faith conjecture depends on the existence of a D-D-bimodule over a division ring D satisfying certain topological conditions. © Glasgow Mathematical Journal Trust 2000.