On retractability and its relationships with lattices associated to a ring

Abstract

The purpose of this work is threefold. First, we explore some relationships between retractability and some lattices of classes of modules. Secondly, we weaken the hypothesis of a result of Ohtake characterizing rings over which all radicals are left exact. In the last section of this work, we introduce a binary relation between modules that produce a Galois connection between the lattice of natural classes and the lattice of conatural classes, and we obtain some results about it.