I came across the following statement in my course notes: 

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> Consider $R = \left\{f \in \mathbb{Q}[X] \mid f(\mathbb{Z}) \subset \mathbb{Z}\right\}$ the *ring of integer-valued polynomials*. Then the ideal generated by $X$ is **not** a prime ideal.
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Isn't this incorrect? I'd argue that $R/(X)$ (where $(X)$ denotes the ideal generated by $X$) is isomorphic to $\mathbb{Z}$, which is an integral domain and thus $(X)$ must be prime.