# Examples of prime ideals that are not maximal

I would like to know of some examples of a prime ideal that is not maximal in some commutative ring with unity.

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sub-rings of $\mathbb{Z}$ and $\mathbb{Z} \times \mathbb{Z}$ and sub-rings in $\mathbb{Z}_n$, but I never thought to check $\{0\}$ – tmpys Feb 28 '14 at 9:15
Would still love some more interesting ones.... – tmpys Feb 28 '14 at 9:15
yes I think $k \mathbb{Z}$ for $k\in\mathbb{Z}$ is an ideal. Am I wrong? I am confused by your question;why would I say it if I did not think it...If you can't help, please just don't comment. Or, in the vein of your question,do you think you are helping with your questions? – tmpys Feb 28 '14 at 9:38
@tmpys You do appear to be confused about the difference between subrings and ideals, and surely it is not unhelpful to point this out to you? (Although I agree that the sentence "I can't confusing you more than you are" is confusing.) – Derek Holt Feb 28 '14 at 10:27
Another example is $(x^2-y)$ in $\mathbb C[x,y]$, see here. – Martin Sleziak Apr 24 at 5:20

Let $R$ be an integral domain and consider $R[x]/(x) \cong R$. It's not a field (unless $R$ is), so $(x)$ is not maximal. Since $R$ has no zero divisors, $(x)$ is a prime ideal.
Note here that $R$ could itself be a suitable polynomial ring, say $K[y]$ for some suitable $K$. – Mark Bennet Feb 28 '14 at 9:53
Take $(0)$, the zero ideal, in $\mathbb{Z}$, which is prime as the integers are an integral domain, but not maximal as it is contained in any other ideal.
The same example works if $\mathbb Z$ is replaced by any other integral domain, which is not a field. – Martin Sleziak Apr 10 '14 at 7:12