Torsion-free Modules over Commutative Domains of Krull Dimension One

Let R be a domain of Krull dimension one, we study when the class F of modules over R that are arbitrary direct sums of finitely generated torsion-free modules is closed under direct summands. If R is local, we show that F is closed under direct summands if and only if any indecomposable, finitely generated, torsion-free module has local endomorphism ring. If, in addition, R is noetherian this is equivalent to say that the normalization of R is a local ring. If R is an h-local domain of Krull dimension 1 and FR is closed under direct summands, then the property is inherited by the localizations of R at maximal ideals. Moreover, any localizations of R at a maximal ideal, except maybe one, satisfies that any finitely generated ideal is 2-generated. The converse is true when the domain R is, in addition, integrally closed, or noetherian semilocal or noetherian with module-finite normalization. Finally, over a commutative domain of finite character and with no restriction on the Krull dimension, we show that the isomorphism classes of countable generated modules in F are determined by their genus.