Change of ring and torsion-theoretic injectivity

Abstract

Let τ be a hereditary torsion theory in R-Mod. Then any ring homomorphism γ: R → S induces in S-Mod a torsion theory σ given by the condition that a left S-module is σ-torsion if and only if it is τ-torsion as a left R-module. We show that if γ: R → S is a ring epimorphism and A is a τ-injective left R-module, then AnnA Ker(γ) is σ-injective as a left S-module. As a consequence, we relate τ-injectivity and σ-injectivity, and we give some applications.