Let $F$ be a number field and let $O_F$ be its ring of integer. I am looking for a definition for the "non-congruency" of two ideals of $O_F$ module another, in an attempt to generalize things we had for $\mathbb{Z}$ in number theory to ideals in $O_F$. I want this definition to be in a way that the number of ideals of $O_F$ that are "non-congruent" module an ideal $I\trianglelefteq O_F$ and are relatively prime to it be equal to $$\phi(I):=N(I)\prod_{p|I}(1-\frac{1}{N(p)})$$ similar to the cardinal of the reduced residue system in $\mathbb{Z}$.
I first tried to create something similar to "$a\equiv b$ (mod $m$) iff $m|a-b$". For this, I let "$I\equiv J$ (mod $K$) iff $I+J\subseteq K$". This looks good at first because we know that every ideal in a Dedekind Domain is uniquely factorized into prime ideals and if $\mathrm{a}\subseteq \mathrm{b}$ then for every prime ideal $p$ we have $n_p(\mathrm{a})\geq n_p(\mathrm{b})$. But taking a closer look, this relation fails to be an equivalence relation since $I\not \equiv I$.
I have looked through some books and seen that $\phi(I)$ is often defined to be the number of units in $\frac{O_F}{I}$. With this definition, one can prove the formula given above. But I don't see how "the units in $\frac{O_F}{I}$" can be related to "non-congruency of ideals module another".
Any idea would be appreciated. If you've seen any books or articles that have introduced this concept, please let me know.
This is not a proper answer, just too long for a comment.
It's an interesting question, but I fear you will not find a satisfactory answer. The ideals in a number ring are very good at replicating the multiplicative structure of the integers, but less so for the additive structure. Note that in a PID, the analogue is clear, but when PID fails, the fact that you have multiple generators causes trouble.
The perhaps slightly less naïve idea, and the way to relate it to units, is to think of the image of prime ideals under the quotient map to $O_F/I$ and hope you can say something about the units in the image. The problem is, either they are zero, or they only contain zero divisors, or they are the entire ring. This follows from unique prime ideal factorization. Even for principal ideals, picking a generator and sending it to the quotient will not give you a canonical unit, since $\langle\alpha\rangle = \langle u\alpha\rangle$ for any unit, but $\alpha \not\equiv u\alpha$ mod $I$ if $u-1\notin I$. In general, ideals are bad at capturing any information about units.