Perfect Torsion Theories

Abstract

The purpose of this paper is to introduce the study of perfect torsion theories on $\operatorname {Mod} R$ and to dualize the concept of divisible module. A torsion theory on $\operatorname {Mod} R$ is called perfect if every torsion module has a projective cover. It is shown for such a theory that the class of torsion modules is closed under projective covers if and only if the class of torsion free modules is closed under factor modules. In addition, it is shown that this condition on a perfect torsion theory is equivalent to its idempotent radical being an epiradical. Codivisible covers of modules are also introduced and we are able to show that any module which has a projective cover has a codivisible cover. Codivisible covers are then characterized in terms of the projective cover of the module and the torsion submodule of the kernel of the minimal epimorphism.