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Keywords:
Hereditary torsion theory $\tau$; $\tau$-injective module; $\tau$-exact module; preradical; exact torsion theory; perfect torsion theory
Hereditary torsion theory $\tau$; $\tau$-injective module; $\tau$-exact module; preradical; exact torsion theory; perfect torsion theory
Summary:
Rim and Teply [10] investigated relatively exact modules in connection with the existence of torsionfree covers. In this note we shall study some properties of the lattice $\Cal E_{\tau}(M)$ of submodules of a torsionfree module $M$ consisting of all submodules $N$ of $M$ such that $M/N$ is torsionfree and such that every torsionfree homomorphic image of the relative injective hull of $M/N$ is relatively injective. The results obtained are applied to the study of relatively exact covers of torsionfree modules. As an application we also obtain some new characterizations of perfect torsion theories.
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