**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.



Do real analytic functions on $\mathbb{C}\mathbb{P}^n$ form a Noetherian ring?
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]


