Valuation domains with a maximal immediate extension of finite rank

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 $P=L_0\supset L_1\supset \mathrm{.}\mathrm{.}\mathrm{.}\supset L_m\supseteq 0$ such that $R_{L_j}\mathrm{/}{L}_{j+1}$ is almost maximal for each $j$, $0\le j\le m-1$ and $R_{L_m}$ is maximal if $L_m\mathrm{\ne }0$. Then we suppose that there is an integer $n\ge 1$ 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 $n\le 3$ if $R$ is almost maximal, and the converse holds if $R$ has a maximal immediate extension of rank $\le 2$.