A module-theoretic characterization of S-(w-)Noetherian rings

Let R be a ring and S a multiplicative subset of R. In this paper, we obtain the ACC characterization, Cartan-Eilenberg-Bass theorem and the absolutely pure characterization 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, if and only if any $(S$-$)$absolutely pure module is S-injective. We also characterized S-w-Noetherian rings similarly.