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Keywords:
precover; cover; hereditary torsion theory $\sigma $; $\sigma $-injective module; $\sigma $-exact module; $\sigma $-pure submodule
precover; cover; hereditary torsion theory $\sigma $; $\sigma $-injective module; $\sigma $-exact module; $\sigma $-pure submodule
Summary:
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.
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