Abstract:Recently Rim and Teply  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.
Keywords: precover, cover, hereditary torsion theory $\sigma $, $\sigma $-injective module, $\sigma $-exact module, $\sigma $-pure submodule
AMS Subject Classification: 16D90, 16S90, 18E40