MAIN MODULES AND SOME CHARACTERIZATIONS OF RINGS WITH GLOBAL CONDITIONS ON PRERADICALS

Main injective modules, which determine every left exact preradical, were introduced in a former work. In this paper, we consider those modules which determine every preradical and we call them main modules. We prove that a main module exists if and only if the lattice of preradicals R-pr is a set, and in this case we give a general construction. Some properties of main modules are proven. We also prove some characterizations of rings for which (a) every preradical is left exact, (b) every preradical is idempotent, (c) every preradical is a radical, (d) every preradical is a t-radical, (e) every preradical which is not the identity functor is prime. These characterizations relate to semisimple artinian rings, rings that are a direct product of a finite number of simple rings, left V-rings, simple rings, among others. In order to illustrate the theory introduced in this paper, several examples are provided.