Totally simple modules

Abstract

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.