On S-Injective Modules

Abstract

Let R be a commutative ring with identity, and let S be a multiplicative subset of R. In this paper, we introduce the notion of S-injective modules as a weak version of injective modules. Among other results, we provide an S-version of Baer’s characterization of injective modules. We also present an S-version of Lambek’s characterization of flat modules: an R-module M is S-flat if and only if its character, Hom_ℤ (M, ℚ/ℤ), is an S-injective R-module. As applications, we establish, under certain conditions, S-counterparts of the Cartan–Eilenberg–Bass and Cheatham-Stone characterizations of Noetherian rings.