Precovers and Goldie’s torsion theory

Abstract

Recently, Rim and Teply , using the notion of $\tau $-exact modules, found a necessary condition for the existence of $\tau $-torsionfree covers with respect to a given hereditary torsion theory $\tau $ for the category $R$-mod of all unitary left $R$-modules over an associative ring $R$ with identity. Some relations between $\tau $-torsionfree and $\tau $-exact covers have been investigated in . The purpose of this note is to show that if $\sigma = (\mathcal T_{\sigma },\mathcal F_{\sigma })$ is Goldie’s torsion theory and $\mathcal F_{\sigma }$ is a precover class, then $\mathcal F_{\tau }$ is a precover class whenever $\tau \ge \sigma $. Further, it is shown that $\mathcal F_{\sigma }$ is a cover class if and only if $\sigma $ is of finite type and, in the case of non-singular rings, this is equivalent to the fact that $\mathcal F_{\tau }$ is a cover class for all hereditary torsion theories $\tau \ge \sigma $.