Large indecomposable modules over local rings

For commutative, Noetherian, local ring R of dimension one, we show that, if R is not a homomorphic image of a Dedekind-like ring, then R has indecomposable finitely generated modules that are free of arbitrary rank at each minimal prime. For Cohen–Macaulay ring R, this theorem was proved in [W. Hassler, R. Karr, L. Klingler, R. Wiegand, Indecomposable modules of large rank over Cohen–Macaulay local rings, Trans. Amer. Math. Soc., in press]; in this paper we handle the general case.