FP-Injective, Simple-Injective, and Quasi-Frobenius Rings#

Abstract

Motivated by a result of Mochizuki [Mochizuki, H. (1965). Finitistic global dimension for rings. Pacific J. Math.15(1):249–258] that, for a left perfect ring R with Jacobson radical if and only if R is right self-injective, we prove here that, for a semiperfect ring R with essential right socle S r , (1) R is right FP-injective if every R-homomorphism from a finitely generated small submodule of a free right R-module F to S r can be extended to an R-homomorphism from F to R, (2) R is right simple-injective if every R-homomorphism from a small right ideal of R to R with simple image can be extended to an R-homomorphism from R to R, and (3) R is right self-injective if every R-homomorphism from a small right ideal of R to R can be extended to an R-homomorphism from R to R. As consequences, several known results on right self-injective rings, right FP-injective rings and right simple-injective rings are extended to larger classes of rings.