Valuation domains with a maximal immediate extension of finite rank

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 $\leq 2$.