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Question: Is the ring of real-analytic functions on $\mathbb{C}\mathbb{P}^n$ (real valued) a Noetherian ring?

References or counterexamples are welcome.

I know that the ring of germs of holomorphic functions on $\mathbb{C}^n$ is Noetherian, but the ring of holomorphic functions on $\mathbb{C}^n$ is not ! I also know how to verify that the ring of polynomials is Noetherian.

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Since I found the answer to my 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 Stein 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]

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