Hi everyone, this is my first post. Hope you don't mind that I'm using it to ask a homework question.

Here's the problem:

Let $\displaystyle I = \{f\in\mathbb{Z}\left[x\right] : f(0) = 0\}$.

- Show $\displaystyle I$ is a prime ideal.
- Show $\displaystyle I$ is not maximal.

For 1), I let $\displaystyle f(x),g(x)\in\mathbb{Z}\left[x\right]$ and $\displaystyle f(x)g(x)\in I$; that is, $\displaystyle f(0)g(0) = 0$. Because $\displaystyle \mathbb{Z}\left[x\right]$ is an integral domain, it has no zero divisors; thus, either $\displaystyle f(0)$ or $\displaystyle g(0)$ must be $\displaystyle 0$ and hence in $\displaystyle I$. Is this right?

I wasn't sure about 2); can I take $\displaystyle J = \{f\in\mathbb{Z}\left[x\right] : f(0) = 0$ or $\displaystyle f(\frac{1}{2}) = 0\}$?

Thanks, Sam