On $S$-injective modules

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, $\text{Hom}_{\mathbb{Z}}(M, \mathbb{Q}/\mathbb{Z})$, 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.