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Let $\mathcal{O}$ be the ring of all algebraic integers in $\overline{\Bbb{Q}}$. It is a non-Noetherian integral domain of dimension $1$. For any non-zero prime $\mathfrak{p} \in \Bbb{Z}$, there are infinitely many primes $\mathfrak{p}_i$ in $\mathcal{O}$ lying above $\mathfrak{p}$, and so $\bigcap \mathfrak{p}_i \neq 0$.