Reduction of local uniformization to the case of rank one valuations for rings with zero divisors

TL;DR

This paper extends the reduction of local uniformization to rank one valuations from integral domains to rings with zero divisors and nilpotents, broadening the applicability of previous results.

Contribution

It generalizes the reduction of local uniformization to rank one valuations to include rings with zero divisors and nilpotents, not just integral domains.

Findings

Reduction applies to rings with zero divisors

Extends previous results to non-integral rings

Broadens the scope of local uniformization techniques

Abstract

This is a continuation of a previous paper by the same authors. In the former paper, it was proved that in order to obtain local uniformization for valuations centered on local domains, it is enough to prove it for rank one valuations. In this paper, we extend this result to the case of valuations centered on rings which are not necessarily integral domains and may even contain nilpotents.