In this article, new results on the Gabriel localizations are obtained. As an application of them, it is shown that a morphism of rings is a flat epimorphism of rings if and only if it corresponds to a kind of the Gabriel localizations. Using this result, new progress in the understanding of the structure of flat epimorphisms of rings have been made. Especially among them, a set-theoretical gap in the structure of the ring M(R), the maximal flat epimorphic extension of a ring R, has been fixed.
