On flat epimorphisms of rings and pointwise localizations

TL;DR

This paper investigates flat epimorphisms and pointwise localizations in commutative rings, establishing characterizations of absolutely flat rings and properties of localizations, with new proofs and insights into their structure.

Contribution

It provides new characterizations of absolutely flat rings via pointwise localizations and explores properties of localizations and epimorphisms in commutative rings.

Findings

A ring is absolutely flat iff it is isomorphic to its pointwise localization.

Characterization of the surjectivity of canonical maps to localizations at minimal primes.

A new proof that contraction-extension of ideals equals the same ideal in flat epimorphisms.

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.