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## Main Question or Discussion Point

Let [tex]M [/tex]be a compact manifold and [tex]C(M), C^{\infty}(M)[/tex] denote rings of continuous (resp. smooth) real functions on [tex]M[/tex]. Let m be a maximal ideal of functions vanishing at some point [tex]x_{0} \in M[/tex]. Prove that [tex]m[/tex] is finitely generated over [tex]C^{\infty}(M)[/tex], but is not finitely generated over [tex]C(M)[/tex].