On flat epimorphisms of rings and pointwise localizations

Abstract

In this paper all rings are commutative. We prove some new results on flat epimorphisms of rings and pointwise localizations. Especially among them, it is proved that a ring $R$ is an absolutely flat (von-Neumann regular) ring if and only if it is isomorphic to the pointwise localization $R^{(-1)}R$, or equivalently, each $R-$algebra is $R-$flat. For a given minimal prime ideal $\mathfrak{p}$ of a ring $R$, the surjectivity of the canonical map $R\rightarrow R_{\mathfrak{p}}$ is characterized. Finally, we give a new proof to the fact that in a flat epimorphism of rings, the contraction-extension of an ideal equals the same ideal.