On S-Artinian and S-Noetherian dimensions

Abstract

Let S be a multiplicative subset of a ring R, and let M be an R-module.
In this article, we introduce and study the concepts of S-Artinian dimension and S-Noetherian dimension of an R-module M. These dimensions are ordinal numbers, and in essence, they measure the deviation of an R-module M from being S-Artinian and S-Noetherian. We observe some basic facts for modules with these dimensions, similar to the basic properties of modules with the Krull and Noetheian dimensions.