## Ladislav Bican

*Relatively exact modules *

Comment.Math.Univ.Carolinae 44,4 (2003) 569-576. **Abstract:**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.

**Keywords:** Hereditary torsion theory $\tau $, $\tau $-injective module, $\tau $-exact module, preradical, exact torsion theory, perfect torsion theory

**AMS Subject Classification:** 16S90, 18E40, 16D80

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