Localization in rings of finite uniform dimensions

Abstract

A definition of torsion theory T on the category R-mod of left R-modules is given, using an equivalence relation on the class of all injective left R-modules. For a torsion theory T, the category R-mod/t of t-torsion-free and T -injective modules is shown to be abelian. A necessary and sufficient condition is given under which a ring A has finite left uniform dimension and zero left singular ideal. Let I denote an ideal of a ring R, and sigma the torsion radical cogenerated by the R-injective envelope E(R/I) of R/I. For a left R-module M, we denote Msigma= Qsigma(M), where Qsigma is the quotient functor corresponding to the torsion radicalsigma. The focal point of our dissertation is the following: THEOREM (Beachy). Let I be an ideal of R and let be the torsion radical cogenerated by E(R/I).Then the following conditions are equivalent. (1) (R/I)sigma is a finite direct sum of simple objects in the category of torsion-free and injective modules.(2) The ring R/I has finite left uniform dimension and zero left singular ideal.