On <i>S</i>-Injective Modules

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