Desingularization of regular algebras

The goals of this paper are two-fold. First, motivated by the uniformization theorem of Zariski, we investigate the rings that can be written as a direct limit of noetherian regular rings. Second, as reverse to the ?rst one, we study properties of a direct limit of noetherian regular rings. We establish the regularity of a direct limit of noetherian regular rings in several cases. By using an extended version of a result of Hartshorne, we remark that a coherent super-regular ring dominates a noetherian regular local ring via a flat extension. Many examples are presented to illustrate the idea and to sharpening the results. We give several applications of the results.