Are the rings of the functions $\mathbb{R}\to\mathbb{R}$ and $\mathbb{Z}_n\to\mathbb{Z}_n$ Noetherian?

Question

I want to check which of the following rings are noetherian:

a) the ring of the functions $\mathbb{R}\to\mathbb{R}$.

b) the ring of the functions $\mathbb{Z}_n\to\mathbb{Z}_n$, $n>1$.

This is the first time I see these kind of rings and I don't know how to work with them.

Answer 1

For the first case you can try the ideals generated by the characteristic functions over the sets $(-n,n)$. If I am not wrong that would give you an increasing chain of ideals which not stabilizes. For the second case, someone has given you a hint.