LetRbe a commutative ring and let ℒ(σ) be a Gabriel filter ofRsuch thatRisσ-noetherian. We discuss the decomposition of theσ1-torsion submodule of aσ-torsionfreeR-module and characterize theσl-injectivity ofσ-closedR-modules through theσm-injectivity of modules over noetherian local rings (S, m). As an application, we obtain new criteria to determine injectivity of modules over noetherian rings, of finite Krull dimension, and Krull domains.