Module approximations and direct limits

This master's thesis deals with questions about the existence of module appro- ximations, namely C-precovers and C-covers for a given class C of R-modules, and studies the relations of these approximations with direct limits. Thanks to a the- orem due to Enochs, we know that every R-module has a C-cover if the pre- covering class C is closed under direct limits, although the validity of the con- verse implication remains an open problem known as Enochs' conjecture. In this setting, we show that any module M with perfect decomposition satisfies that the class Add(M) is precovering and closed under direct limits; hence also cove- ring. Furthermore, we prove Enochs' conjecture for Add(M) if M is small, e.g. < ℵω-generated. Specifically, if M is small and Add(M) covering, then M has a perfect decomposition