Let $M$, $N$ be connected nondiscrete compact smooth manifolds. Can the ring of continuous functions on $M$ be isomorphic to the ring of smooth functions on $N$?



Do rings of smooth functions differ from rings of continuous functions?
No. In both the smooth function ring and the continuous function ring a maximal ideal $\frak m$ consists of the functions vanishing at some point. In the smooth case $\frak m/\frak m^2$ is the cotangent space of the manifold at that point, while in the continuous case $\frak m^2=\frak m$.


