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? 



The ring $C^{\infty}(M)$?
You are correct: $C^\infty(M)$ does contain all the geometry and topology
of $M$ (at least when it is considered as an $\mathbb{R}$-algebra).
For example when $M$ is compact the points of $M$ correspond to the maximal
ideals of $C^\infty(M)$ (this is quite easy to prove). If $M$ is not compact
there are maximal ideals $I$ not corresponding to points, but these can be distinguished
since $C^\infty(M)/I$ is a proper extension of $\mathbb{R}$ for such $I$.


See the book [Smooth Manifolds and Observables](http://books.google.co.uk/books?id=N5mHmFiovgkC&printsec=frontcover&dq=Jet+nestruev&source=gbs_similarbooks_s&cad=1#v=onepage&q&f=false) by "Jet Nestruev" to see these
ideas fully worked out.


