Precovers and Goldie's torsion theory

Abstract: Recently, Rim and Teply \cite{8}, 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 \cite{5}. The purpose of this note is to show that if $\sigma= (\Cal T_{\sigma},\Cal F_{\sigma})$ is Goldie's torsion theory and $\Cal F_{\sigma}$ is a precover class, then $\Cal F_{\tau}$ is a precover class whenever $\tau\geq\sigma$. Further, it is shown that $\Cal F_{\sigma}$ 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 $\Cal F_{\tau}$ is a cover class for all hereditary torsion theories $\tau\geq\sigma$.

Keywords: hereditary torsion theory, Goldie's torsion theory, non-singular ring, precover class, cover class

Classification (MSC2000): 16S90, 18E40, 16D80

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