Totally simple modules Jara Martínez, Pascual Omar, Farah Santos, Evangellina Simple module S-finite module Noetherian ring Hereditary torsion theory Simple modules on a ring have information about a part of the spectrum (the maximal spectrum) and, in some cases, about the whole ring. Therefore, knowledge about the structure and properties of simple modules is of interest. In the case we are interested in: chain conditions on modules relative to a multiplicative set S ⊆ A or a hereditary torsion theoryσ inMod- A,wefindthattwodifferentclassesoftotallysimplemodules appear. Given a multiplicative subset S ⊆ A one tends to introduce S-simple modules either as those non totally S-torsion which are S-minimal, or as those for which 0 ⊆ M is S-maximal. Apparently these two definitions are different. We show that both definitions coincide, and define an A-module M to be S-simple whenever it satisfies: (1) Ann(M) ∩ S = ∅; (2) there exists s ∈ S such that σS(M)s = 0, and (3) Ms ⊆ L, for every no totally S-torsion submodule L ⊆ M. The main goal of this paper is to provide examples of this kind of totally simple modules by delving into their structure. As a byproduct we explore the relationship between these totally simple modules and totally prime modules, and the local behaviour of totally simple modules. We complete the paper by providing examples of this theory. 2025-01-21T08:32:58Z 2025-01-21T08:32:58Z 2024 journal article Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry https://hdl.handle.net/10481/99788 10.1007/s13366-024-00759-6 eng embargoed access Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry