6 - Simple Injective and Dual Rings

Summary

We now introduce another class of right mininjective rings that seems to effectively mirror the properties of right self-injective rings. A ring R is called right simple injective if every R-linear map with simple image from a right ideal to R extends to R. Examples include right mininjective rings and right self-injective rings, but there are simple injective rings that are not right P-injective, and there exist right simple injective rings that are not left simple injective. If R is right simple injective, so also is eRe, where e is an idempotent satisfying ReR = R, but simple injectivity is not a Morita invariant [we characterize when M_n(R) is right simple injective].

We show that a simple injective ring R is right Kasch if and only if every right ideal is an annihilator and, in this case, that R is left P-injective, soc(eR) is simple and essential in eR for every local idempotent e in R, and R is left finite dimensional if and only if it is semilocal. In fact, if R is semiperfect, right simple injective and soc(eR) ≠ 0 for every local idempotent e (the analogue of the right minfull rings), then R is a right and left Kasch, right and left finitely cogenerated, right continuous ring in which S_r = S_l (= S) and both soc(eR) = eS and soc(Re) = Se are simple and essential in eR and Re, respectively, for every local idempotent e in R.