# What's stronger: projective or locally free? flat or locally free?

maybe that's an idiot question, however I did not found anything related in the classical references. It's know that a finitely generated projective $A$-module $M$ is locally free ,since each localization $A_\mathfrak{p}$ is a local ring (then the result follows by a theorem of Kaplansky). Moreover, if $A$ is Noetherian and $M$ finitely generated, then flatness and locally freeness coincide. However I don't know exactly if these results holds for arbitrary modules. Furthermore, I don't know the relation between flatness and locally freeness for general rings and modules? Are there any results for such general facts? Any references? How one can (partial) order these properties (flatness, locally freeness, "projectiveness") when assuming certain common classes of rings if necessary? Is it possible to (partial) order these properties for general abelian categories (making some restrictions if necessary)?