Spectral torsion theories in module categories

Abstract

Let M be an R-module and α[M] the category of all R-modules that are subgenerated by M. For any hereditary torsion theory T in α[M], the class of quotient modules defines a Grothendieck category denoted by ∊_T[M]. T is called spectral if ∊_T[M] is a spectral category (every short exact sequence in∊_T[M] splits). The paper is devoted to the investigation of such torsion theories. Various characterizations are obtained and properties of the endomorphism rings of the corresponding quotient modules are studied. The final sectlon is concerned with the role of simple and semisimple modules with respect to spectral torsion theories. Our presentation generalizes and extends related results for spectral torsion theories in the category of all R-modules R-Mod.