9 - A Generic Example

Summary

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.

Generalities

If S is any ring and _SV_S, _SW_S, and _SP_S 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 _DV_D and _DP_D be nonzero bimodules over a division ring D, and suppose a bimap V × V → P is given.