Prime $\ell \in \mathbb{Z}$ where $\ell\mathcal{R}_i$ prime ideal of $\mathcal{R}_i$ for all $i = 1, 2, \ldots, n$? (Exercise 5.5, Silverman)

Question

Let $\mathcal{K}/\mathbb{Q}$ be an imaginary quadratic field, and let $\mathcal{R}_1, \ldots, \mathcal{R}_n$ be orders in $\mathcal{K}$. Is there a prime $\ell \in \mathbb{Z}$ such that $\ell\mathcal{R}_i$ is a prime ideal of $\mathcal{R}_i$ for all $i = 1, 2, \ldots, n$?

Note: this is exercise 5.5, Silverman "Arithmetic of elliptic curves".

Answer 1

There are only finitely many primes dividing the $\mathcal{R}_i$, so choose any inert prime of $\Bbb Q$ not in this list--one exists because there are infinitely many cf. the Chebotarev density theorem and the fact that only finitely many primes ramify. Then $\ell \mathcal{R}_i$ is prime by the usual isomorphism theorems.