Indecomposable modules over Nagata valuation domains

Abstract

For a discrete valuation ring R, let fr ( R ) {\text {fr}}(R) be the supremum of the ranks of indecomposable finite rank torsion-free R-modules. Then fr ( R ) = 1 , 2 , 3 {\text {fr}}(R) = 1,2,3 , or ∞ \infty . A complete list of indecomposables is given if fr ( R ) ≤ 3 {\text {fr}}(R) \leq 3 , in which case R is known to be a Nagata valuation domain.