Is there an ideal of $(C^0(\mathbb{R}),+,\cdot)$ that isn't prime?

Question

Let $C^o(\mathbb{R})$ be the ring of continuous functions over $\mathbb{R}$ with multiplication and addition of functions.

What sort of ideals can exist in that ring apart from those that are caracterized by a root (for example all the funtions $f$ s.t. $f(3)=0$)

Is there an ideal that isn't prime?

I found the $I=\{f\in C^o(\mathbb{R})\ :\ f(x_1)=0$ and $f(x_2)=0\}$ if we take $f(x)=x-x_1$ and $g(x)=x-x_2$ we have that $fg\in I$ but $f\notin I$ and $g\notin I$

There is also $\langle p(x) \rangle$ where $p$ is a recucible polynomial in a ring of polynomials

Are there different examples?

Answer 1

The zero ideal isn't prime, for example.

It's a well-known exercise (for example, Kaplansky's Commutative rings page 7 exercise #1, the first in the book) that for a commutative ring, having all proper ideals prime is equivalent to being a field.