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 ⊃ L 1 ⊃ ⋯ ⊃ L m ⊇ 0 such that R L j / L j + 1 is almost maximal for each j, 0 ⩽ j ⩽ m − 1 and R L m is maximal if L m ≠ 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.
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