kt.k theory homology - The ring $C^{infty}(M)$?

Let $M$ be a smooth paracompact manifold. I think that the ring $C^{infty}(M)$ contains many (possibly almost all?) geometric or topological information about $M$.

(e.g. Let $E$ be a vector bundle over $M$,$Gamma(E)$ be a set of smooth section of $E$. Then, $Gamma(E)$ is a $C^infty(M)$-module. (Actually, I think $Gamma(E)$ is projective $C^infty(M)$-module because every a short exact sequence of vector bundle splits.))

But I have a feeling that $C^infty(M)$ is too large to change the problem of Manifold theory into an algebraic problem or Ring theoretic problem.

Are there any well-known concrete description about the ring $C^infty(M)$ for some manifold $M$ with simple topology?

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