S-Noetherian rings, modules and their generalizations

Let R be a commutative ring with identity, M an R-module and S ⊆ R a multiplicative set. Then M is called S-finite if there exist an s ∈ S and a finitely generated submodule N of M such that sM ⊆ N. Also, M is called S-Noetherian if each submodule of M is S-finite. A ring R is called S-Noetherian if it is S-Noetherian as an R-module. This paper surveys the most recent developments in describing the structural properties of S-Noetherian rings, S-Noetherian modules and their generalizations. Some interesting constructed examples of S-Noetherian rings and modules are also presented.