Covers, Preenvelopes, and Purity

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 Saorín. 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. 0. Introduction Covers. The main topic of this paper is the notion of covering classes. To explain what that means, observe that the classical ho-mological algebra of a ring can be phrased in terms of the class of projective modules. This class permits the construction of projective resolutions which again enable the computation of derived functors. In relative homological algebra, the class of projective modules is replaced by another, suitably chosen class of modules. This replaces projective resolutions by resolutions in terms of modules in the chosen class, and derived functors by relative derived functors. A classical example of this is pure homological algebra where the projective modules are replaced by the so-called pure projective modules; these are the direct summands in direct sums of finitely presented modules. Pure homological algebra is a useful tool with a number of applications; see for instance [12].