Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 136 / 1 / b / i / Solution

The ring $\mathbb{Z}[T]$ is Noetherian and integrally closed, but it is not a Dedekind domain because it has Krull dimension two. Concretely, the nonzero prime ideal $(2)$ is properly contained in the prime ideal $(2,T)$ and is therefore not maximal.