This is a follow-up to this question: Localization of finite modules, or: compatibility of ideal norms with localization at a prime number
Let $K$ be an algebraic number field and $N:\mathcal{I}(\mathcal{O}_K)\to \mathbb{Z}$ be $N(\mathfrak{a})=\# (\mathcal{O}_K/\mathfrak{a})$, where $\mathcal{I}(\mathcal{O}_K)$ denotes the set of non-zero ideals of $\mathcal{O}_K$.
Let $S\subset \mathcal{O}_K$ be a multiplicative subset. Is there a way to make sense of a "norm of ideals" in $S^{-1}\mathcal{O}_K$ that is "compatible" with the norm of $\mathcal{O}_K$?
I'm especially interested in the case $S=\mathbb{Z}\setminus p \mathbb{Z}$, where $p$ is a prime number.
In the previous question the answer by froggie immediately shows that we can't naively define it (in this special case) as the cardinality of the quotient.
I know the question is vaguely phrased, but I seem to recall reading about such a thing, and I can't remember where.