Decomposition and classification of length functions

TL;DR

This paper explores how length functions on integral domains can be decomposed into sums derived from overrings, providing standard representations under specific conditions and linking singular length functions to localizing systems.

Contribution

It introduces a method to decompose length functions on integral domains and establishes a correspondence between singular length functions and localizing systems.

Findings

Standard representation when the domain admits a Jaffard family

Decomposition results for Noetherian and Prüfer domains

Bijective correspondence between singular length functions and localizing systems

Abstract

We study decompositions of length functions on integral domains as sums of length functions constructed from overrings. We find a standard representation when the integral domain admits a Jaffard family, when it is Noetherian and when it is a Pr\"ufer domains such that every ideal has only finitely many minimal primes. We also show that there is a natural bijective correspondence between singular length functions and localizing systems.