Rings of analytic functions definable in o-minimal structure

From the ring theoretical viewpoint, especially from the viewpoint of Real Algebra, we consider the ring of analytic functions definable in a given o-minimal expansion of the real field on a definable real analytic manifold. We find necessary conditions for o-minimal structures that Artin–Lang property, Real Nullstellensatz and Hilbert 17th Problem for this ring hold true in the three-dimensional case. We also prove that this ring is Noetherian in the three-dimensional case when the given o-minimal structure is the restricted analytic field.