Every maximal ideal is prime... why not converse? I know that every maximal ideal is prime but I don't see why the converse doesn't hold.
Intuitively it seems like every prime ideal should be maximal. Off the top of my head I can't imagine how we could have an ideal that would 'contain' a prime ideal. 

If $R$ is any integral domain which is not a field, then $(0)$ is a prime ideal in $R$, but is not a maximal ideal. For example, $R=\mathbb{Z}$ is an integral domain which is not a field.
