Elementary varieties and existence of flat covers

It is shown that flat covers exist in a wide class of additive categories – we call them elementary – which behave similar to locally finitely presented Grothendieck categories. Elementary categories have enough “finitely presented” objects, but they need not be locally finitely presented. This is related to the existence of pure monomorphisms which are not kernels and the non-exactness of direct limits. For a module category Mod(R), every class of R-modules containing R cogenerates an elementary full subcategory.