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Feb 5 at 11:56 comment added Thomas Kurbach Clever! Your argument in the smooth case definitely shows that finitely generated $C^0$-closed ideals in $C^{\infty}$ are of the form $J_X$ for some $X$. I will check the link you provided, but the same strategy will of course not work in the real analytic setting...at least not directly.
Feb 5 at 0:59 history answered Igor Khavkine CC BY-SA 4.0