Cotorsion radicals and projective modules

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<jats:p>We study the notion (for categories of modules) dual to that of torsion radical and its connections with projective modules.</jats:p><jats:p>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 <jats:italic>R<jats:sup>M</jats:sup></jats:italic> correspond to the idempotent ideals of <jats:italic>R</jats:italic>. Thus they also correspond to TTF classes in the sense of Jans [9].</jats:p><jats:p>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 <jats:italic>S</jats:italic>-ring in the sense of Kasch. This extends the similar result of Morita [15] for artinian rings.</jats:p>

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