Precovers

summary:Let $\mathcal G$ be an abstract class (closed under isomorpic copies) of left $R$-modules. In the first part of the paper some sufficient conditions under which $\mathcal G$ is a precover class are given. The next section studies the $\mathcal G$-precovers which are $\mathcal G$-covers. In the final part the results obtained are applied to the hereditary torsion theories on the category on left $R$-modules. Especially, several sufficient conditions for the existence of $\sigma $-torsionfree and $\sigma $-torsionfree $\sigma $-injective covers are presented