Geodesic
Relative exact covers
Commentationes Mathematicae Universitatis Carolinae, Tome 42 (2001) no. 4, pp. 601-607.

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Recently Rim and Teply [11] found a necessary condition for the existence of $\sigma$-torsionfree covers with respect to a given hereditary torsion theory for the category $R$-mod. This condition uses the class of $\sigma$-exact modules; i.e. the $\sigma$-torsionfree modules for which every its $\sigma$-torsionfree homomorphic image is $\sigma$-injective. In this note we shall show that the existence of $\sigma$-torsionfree covers implies the existence of $\sigma$-exact covers, and we shall investigate some sufficient conditions for the converse.
Classification : 16D50, 16D90, 16S90, 18E40
Keywords: precover; cover; hereditary torsion theory $\sigma $; $\sigma $-injective module; $\sigma $-exact module; $\sigma $-pure submodule 
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Bican, Ladislav; Torrecillas, Blas. Relative exact covers. Commentationes Mathematicae Universitatis Carolinae, Tome 42 (2001) no. 4, pp. 601-607. http://geodesic.mathdoc.fr/item/CMUC_2001__42_4_a0/