$\mathbb{Z}[\theta]$ can differ from the ring of integers

There exist a monic polynomial $f \in \mathbb{Z}[x]$, irreducible in $\mathbb{Z}[x]$, and a complex root $\theta$ of $f$, together with an element $x \in \mathbb{Q}(\theta)$ such that

1. $x$ is an algebraic integer, i.e. a root of a monic polynomial with integer coefficients, and
2. $x \notin \mathbb{Z}[\theta]$, the subring of $\mathbb{C}$ generated by $\theta$.

So the ring of integers $\mathcal{O}_{\mathbb{Q}(\theta)}$ can strictly contain $\mathbb{Z}[\theta]$. (A standard example is $\theta = \sqrt{5}$, $x = (1+\sqrt5)/2$.)