This is an obvious continuation of [an MO question](https://mathoverflow.net/q/431642). Let $r,s\in\mathbb N\cup\{\infty\}$ with $r\neq s$, and $M,N$ two connected manifolds of positive dimension (which roots out the trivial case of a single point). I wonder whether the rings $C^r(M)$ and $C^s(N)$ of real-valued continuously $r$-th (resp. $s$-th) differentiable functions are non-isomorphic?

More precisely, we have the following (combinations of) variants of conditions on $M$ and $N$, although they might not affect the result nontrivially.

1. Smoothness:

* weakest: $M$ is a $C^r$-manifold and $N$ is a $C^s$-manifold;
* strongest: both $M$ and $N$ are $C^\infty$.

2. Compactness:

* weakest: both are paracompact Hausdorff (usually present in the definition of manifolds);
* strongest: both are compact Hausdorff.

There is also a variant of manifolds with boundaries, but I am not looking at this generality in this question.

It seems that, in this case, Morse's lemma might not hold, thus the structure of vanishing ideals $\mathfrak m\_x=\ker(C^r(M)\to\mathbb R)$ at a point $x\in M$ might be more complicated (say, maybe not regular). However, it seems possible that [Tom Goodwillie's answer](https://mathoverflow.net/a/431644/) applies to the situation that one of $r$ and $s$ is $\infty$.