S-Noetherian rings and their extensions

Let R be an associative ring with identity, S a multiplicative subset of R, and M a right R-module. Then M is called an S-Noetherian module if for each submodule N of M, there exist an element s ∈ S and a finitely generated submodule F of M such that Ns ⊆ F ⊆ N, and R is called a right S-Noetherian ring if R_R is an S-Noetherian module. In this paper, we study some properties of right S-Noetherian rings and S-Noetherian modules. Among other things, we study Ore extensions, skew- Laurent polynomial ring extensions, and power series ring extensions of S-Noetherian rings.