Valuation domains with a maximal immediate extension of finite rank

TL;DR

This paper investigates valuation domains with finite rank maximal immediate extensions, establishing structural properties and bounds on torsion-free module decompositions, revealing a connection between the domain's maximal immediate extension rank and module decomposition limits.

Contribution

It proves the existence of a prime ideal sequence characterizing valuation domains with finite rank immediate extensions and bounds the decomposition rank of torsion-free modules based on the domain's properties.

Findings

Existence of a prime ideal sequence with almost maximal quotients

Bound n ≤ 3 for torsion-free modules in almost maximal domains

Converse: if immediate extension rank ≤ 2, then n can be 3

Abstract

If $R$ is a valuation domain of maximal ideal $P$ with a maximal immediate extension of finite rank it is proven that there exists a finite sequence of prime ideals $P=L_0\supset L_1\supset...\supset L_m\supseteq 0$ such that $R_{L_j}/L_{j+1}$ is almost maximal for each $j$, $0\leq j\leq m-1$ and $R_{L_m}$ is maximal if $L_m\ne 0$. Then we suppose that there is an integer $n\geq 1$ such that each torsion-free $R$-module of finite rank is a direct sum of modules of rank at most $n$. By adapting Lady's methods, it is shown that $n\leq 3$ if $R$ is almost maximal, and the converse holds if $R$ has a maximal immediate extension of rank $\leq 2$.