Suppose $R$ is an integral domain and $r\in R$. I am trying to prove that there is a prime ideal of $R$ that does not contain $r$, but am having trouble figuring out how to proceed.
If $r$ is nonzero, then we can take the ideal $\{0\}$, which is prime since $R$ has no zero divisors.
So suppose that $r=0$. But every ideal contains $0$. What am I missing?