  We prove a tight connection between reflexive modules over a one-dimensional
ring $R$ and its birational extensions that are self-dual as $R$-modules.
Consequently, we show that a complete local reduced Arf ring has finitely many
indecomposable reflexive modules up to isomorphism, which can be represented
precisely by the local rings infinitely near it. We also characterize Arf rings
by the property that any reflexive module is self-dual. We give applications on
dimension of subcategories and existence of endomorphism rings of small global
dimension (non-commutative desingularizations). Our results indicate striking
similarities between Arf rings in dimension one and rational singularities in
dimension two from representation-theoretic and categorical perspectives.
