Since I found the answer to may question I will write it here, hoping that it will be helpful to someone else.
Answer: The ring of global real analytic functions on a compact real analytic manifold is noetherian due to Theorem I.9. [J.Frisch.: Points de platitude d'un morphisme d'espaces analytiques complexes. Invent. Math., 4:118-138, 1967.], see also last paragraph in [J. M. Ruiz. On Hilbert's 17th problem and real Nullstellensatz for global analytic functions. Math. Z., 190(3):447-454, 1985]