Cotorsion radicals and projective modules

Abstract

We study the notion (for categories of modules) dual to that of torsion radical and its connections with projective modules.Torsion radicals in categories of modules have been studied extensively in connection with quotient categories and rings of quotients. (See [8], [12] and [13].) In this paper we consider the dual notion, which we have called a cotorsion radical. We show that the cotorsion radicals of the category RM correspond to the idempotent ideals of R. Thus they also correspond to TTF classes in the sense of Jans [9].It is well-known that the trace ideal of a projective module is idempotent. We show that this is in fact a consequence of the natural way in which every projective module determines a cotorsion radical. As an application of these techniques we study a question raised by Endo [7], to characterize rings with the property that every finitely generated, projective and faithful left module is completely faithful. We prove that for a left perfect ring this is equivalent to being an S-ring in the sense of Kasch. This extends the similar result of Morita [15] for artinian rings.