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=L0⊃L1⊃⋯⊃Lm⊇0 such that RLj/Lj+1 is almost maximal for each j, 0⩽j⩽m−1 and RLm is maximal if Lm≠0. Then we suppose that there is an integer n⩾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⩽3 if R is almost maximal, and the converse holds if R has a maximal immediate extension of rank ⩽2.