The smallest closed subcategory containing the μcomplemented modules

Abstract

Given a hereditary torsion functor , the class of μ-complemented modules was recently introduced by P.F. Smith and the authors as an analogue of extending modules. This current article explores this class by viewing it as a subclass of EX, the smallest closed subcategory which contains it. As a consequence the class of μ-complemented modules is shown here to be closed under the formation of module of quotients. As to EX, we prove that it is closed under arbitrary direct products in Mod-Rif Ris a valuation ring. On the other hand, if Ris commutative Noetherian and μ is jansian then every μ-complemented module is a direct sum of a μ-torsion module and a semisimple, which prompted us to analyze when E Xcontains a subgenerator of this form.