As it is more or less well-know, and as it has come up on MO a couple of times, the $\mathbb R$-algebra $C^\infty(M)$ of smooth functions on a (say) compact manifold contains essentially everything there is to know about $M$ itself.



> 
> Does one *really* need to know the $\mathbb R$-algebra structure, though? Can we manage with $C^\infty(M)$ given as an abstract ring?
> 
> 
> 


For comparison, if I recall correctly one can reconstruct (as an analytic manifold) an open subset $U\subseteq\mathbb C$ from the field of meromorphic functions on it, given as an abstract field. (It follows, of course, that the ring of holomorphic functions also can be used to reconstruct $U$.)



On the $\mathbb R$-algebra structure on $C^\infty(M)$.
If $M$ is connected then one can determine $\mathbb{R}$ 
(the constant functions) within
$C^\infty(M)$ (the ring of smooth real-valued functions on $M$).
One can certainly determine $\mathbb{Q}$ within $C^\infty(M)$.
If $f$ is not in $\mathbb{Q}$ then it's a constant function iff
$f-a$ is a unit in $C^\infty(M)$ for all $a\in\mathbb{Q}$. The key
is that all nonconstant functions must take a rational value.


This doesn't work for non-connected $M$, since a function $f$
may take distinct constant values on a clopen partition on $M$.
Maybe one can use idempotents...


