5 - Principally Injective and FP Rings

Summary

A ring R is right mininjective if R-linear maps aR → R extend to R_R → R_R whenever aR is a simple right ideal. It is natural to enquire about the rings for which this condition is satisfied for all principal right ideals aR (for example if R is regular or right self-injective). These rings, called right principally injective (or right P-injective), play a central role in injectivity theory. An example is given of a right P-injective ring that is not left P-injective. If R is right P-injective it is shown that Z_r = J, that R is directly finite if and only if monomorphisms R_R → R_R are epic, and that R has the ACC on right annihilators if and only if it is left artinian. If R is right P-injective and right Kasch then S_ι ⊆ ^ess_RR. A semiperfect, right P-injective ring R in which S_r ⊆^ess R_R is called a right GPF ring. Hence the right PF rings are precisely the right self-injective, right GPF rings, and these right GPF rings exhibit many of the properties of the right PF rings: They admit a Nakayama permutation, they are right and left kasch, they are left finitely cogenerated, S_r = S_ι is essential on both sides, and Z_r = J = Z_ι.

Unlike mininjectivity, being right P-injective is not a Morita invariant. In fact, M_n(R) is right P-injective implies that R is right n-injective.