Covers, preenvelopes, and purity

Abstract

We show that if a class of modules is closed under pure quotients, then it is precovering if and only if it is covering, and this happens if and only if it is closed under direct sums. This is inspired by a dual result by Rada and Saor\'{\i}n. We also show that if a class of modules contains the ground ring and is closed under extensions, direct sums, pure submodules, and pure quotients, then it forms the first half of a so-called perfect cotorsion pair as introduced by Salce; this is stronger than being covering. Some applications are given to concrete classes of modules such as kernels of homological functors and torsion free modules in a torsion pair.