PRECOVERS AND GOLDIE’S TORSION THEORY

Abstract. Recently, Rim and Teply [8], using the notion of τ-exact modules, found a nec-essary condition for the existence of τ-torsionfree covers with respect to a given hereditary torsion theory τ for the category R-mod of all unitary left R-modules over an associative ring R with identity. Some relations between τ-torsionfree and τ-exact covers have been investigated in [5]. The purpose of this note is to show that if σ = (Tσ,Fσ) is Goldie’s torsion theory and Fσ is a precover class, then Fτ is a precover class whenever τ> σ. Further, it is shown that Fσ is a cover class if and only if σ is of finite type and, in the case of non-singular rings, this is equivalent to the fact that Fτ is a cover class for all hereditary torsion theories τ> σ