## Ladislav Bican, Blas Torrecillas Relative exact covers

Comment.Math.Univ.Carolinae 42,4 (2001) 601-607.

Abstract: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.

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