The Faith conjecture asserts that every left or right perfect, right self-injective ring R is quasi-Frobenius. The conjecture remains open for semiprimary, local, right self-injective rings with J3 = 0. It is known (Theorem 3.40) that the conjecture is true if J2 = 0. In this section we construct a local ring R with J3 = 0 and characterize when R is artinian or self-injective in terms of conditions on a bilinear mapping from a (D, D)-bimodule to a division ring D ≅ R/J. We conclude by characterizing other properties of R in a similar way.
If S is any ring and SVS, SWS, and SPS are bimodules, a function V × W → P, which we write multiplicatively as (υ,ω) ↦ υω, will be called a bimap if the conditions
(1) (υ + υ1)ω = υω + υ1ω and (Sυ)ω = S(υω),
(2) υ(ω + ω1) = υω + υω1 and υ(ωS) = (υω)S, and
(3) (υS)ω = υ(Sω)
hold for all υ, υ1 in V, all ω, ω1 in W, and all s in S. Our interest is in the case when S = D is a division ring. We construct the following ring, which is the central topic of this chapter.
Definition. Let DVD and DPD be nonzero bimodules over a division ring D, and suppose a bimap V × V → P is given.