A NOTE ON ℵ₀-INJECTIVE RINGS

Abstract

A ring R is called right ℵ 0 -injective if every right homomorphism from a countably generated right ideal of R to R R can be extended to a homomorphism from R R to R R . In this note, some characterizations of ℵ 0 -injective rings are given. It is proved that if R is semiperfect, then R is right ℵ 0injective if and only if every homomorphism from a countably generated small right ideal of R to R R can be extended to one from R R to R R . It is also shown that if R is right noetherian and left ℵ 0 -injective, then R is QF. This result can be looked as an approach to the Faith-Menal conjecture.