[Math] Examples of prime ideals that are not maximal

Question

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

Answer 1

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.