Some characterizations of totally Artinian rings

Abstract

In this paper we attempt to generalize some classical results from Artinian/Noetherian ring theory to hereditary torsion theories. It is worthwhile noticing that the investigation of Artinian and Noetherian rings is an abundant source of findings, primarily related to the structure of rings and modules. By extending the concept of S-Noetherian rings, we explore totally Artinian and totally Noetherian rings. The primary goal of this note is to establish connections between totally Artinian and totally Noetherian rings, and to characterize the former, initially through localization at maximal ideals and later, as a result, by demonstrating that the ring A is totally Artinian if, and only if, it has an Artinian homomorphic image with a totally torsion kernel. The theory is further developed by addressing several open problems and presenting illustrative examples. The central aim of this paper is to offer examples of these types of rings by exploring their underlying structure.