 Solution

In the polynomial ring $\mathbb{Z}[x]$, the ideal $(x)$ is nonzero and prime: the quotient ring $\mathbb{Z}[x]/(x)$ is isomorphic to the integral domain $\mathbb{Z}$, by evaluation at zero. It is not maximal, since that quotient is not a field. Equivalently $(x)⊊(2,x)⊊\mathbb{Z}[x]$. Thus $(x)$ is a nonzero prime ideal which is not maximal.