TITLE: Are $C(\mathbb{R})$ and $D(\mathbb{R})$ isomorphic or not?
QUESTION [7 upvotes]: Let's denote ring of all continuous functions and differentiable functions from $\mathbb{R}$ to $\mathbb{R}$ by $C(\mathbb{R})$ and $D(\mathbb{R})$, respectively. I want to know whether these rings are isomorphic or not.

From the first part of this answer it is clear that if there exists a ring monomorphism which sends 1 to itself then it is identity on the constant functions. But I can not find anything else. Please help. Thank you.

REPLY [6 votes]: Suppose they are isomorphic. Then consider the identity function $Id$ which belongs to $D(\mathbb{R})$. Let the pre image of $Id$ is $f$ in $C(\mathbb{R})$.
Now if $f\in C(\mathbb{R})$ then $f^{1/3}\in C(\mathbb{R})$. Then by properties of isomorphism image of  $f^{1/3}$ is $Id^{1/3}$. But $Id^{1/3}$ is not differentiable at $0$ and so it is not in $D(\mathbb{R})$. Hence contradiction!!