Do the ring of smooth functions on the set of real numbers $\Bbb R$ with the usual pointwise addition and multiplication form an integral domain?
I have been trying to prove this result without any success.Can anyone please help me?Thanks in advance for any help.
If you could construct two smooth functions $f,g:\Bbb R\to\Bbb R$ such that \begin{align*} f(x)&=0\Leftrightarrow x\notin (0,1) \\ g(x)&=0\Leftrightarrow x\notin (2,3) \end{align*} then $f$ and $g$ would be two nonzero elements whose product is zero. The problem now is to find two such functions. Here's a hint on how to do so:
Can you show that the function $\varphi:\Bbb R\to\Bbb R$ given by $$ \varphi(x)= \begin{cases} \displaystyle\exp\left(\frac{1}{x^2-1}\right) & -1<x<1 \\ 0 & \text{otherwise} \end{cases} $$ is smooth and nonzero only on the interval $(-1,1)$? Given that this function has this property, can you use it to define $f$ and $g$?