A module-theoretic characterization of $S$-Noetherian rings

Abstract

Let $R$ be a ring and $S$ a multiplicative subset of $R$. In this note, we obtain the ACC characterization and Cartan-Eilenberg-Bass theorem for $S$-Noetherian rings. In details, we show that a ring $R$ is an $S$-Noetherian ring if and only if any ascending chain of ideals of $R$ is $S$-stationary, if and only if any direct sum of injective modules is $S$-injective, if and only if any direct limit of injective modules is $S$-injective.