p$ be a “small” prime, and let ${\mathbb{A}}_{{\mathbb{Q}},\ell}$ be the restricted product over all $v \ne \ell$. Let ${\mathbb{A}}_B = B \otimes_{\mathbb{Q}}{\mathbb{A}}_{\mathbb{Q}}$ be the adèle ring of $B$, and ${\mathbb{A}}_{B,\ell} = B \otimes {\mathbb{A}}_{{\mathbb{Q}},\ell}$. Then $B$ embeds diagonally in ${\mathbb{A}}_B$ and is discrete in ${\mathbb{A}}_B$ (see [@Cassels1967 Section 14]). The strong approximation theorem (see [@Cassels1967 Section 15]) asserts that $B$ is dense in ${\mathbb{A}}_{B,\ell}$ (see also Théorème Fondamental 1.4, p. 61 of Vignéras [@Vigneras1980]).

The strong approximation theorem can be viewed as a strong version of the Chinese remainder theorem. We apply this to find an element of a left ${\mathcal{O}}$-ideal $I$ which generates $I$ almost everywhere. Each such ideal is known to be generated by two elements $N$ and $\alpha$, where we may take $N = {\mathrm{Nrd}}(I)$ for the first generator. This follows since locally ${\mathcal{O}}_v = {\mathcal{O}}\otimes {\mathbb{Z}}_v$ is a left principal ideal ring, hence so is the quotient ${\mathcal{O}}/N{\mathcal{O}}$. If $I = {\mathcal{O}}(N,\alpha):={\mathcal{O}}N+{\mathcal{O}}\alpha$, the approximation theorem implies that we can find $\beta$ in $I$ such that $$\beta \equiv \alpha \bmod N{\mathcal{O}}$$ and ${\mathrm{Nrd}}(\beta) = N\ell^e$ for some positive integer $e$, from which $I = {\mathcal{O}}(N,\alpha) = {\mathcal{O}}(N,\beta)$. By Lemma \[lem:ideal\_norm\_rep\], an effective version of this strong approximation 