Hello, Let $K/{\mathbb Q}$ be a finite extension which is not necessarily Galois, and ${\mathcal O}$ be the ring of integers of $K$. Let $p$ be a prime in ${\mathbb Q}$ and let $p {\mathcal O}={\mathfrak p}_1^{k_1} \cdots {\mathfrak p}_r^{k_r}$ be its prime factorization in $K$. Next let ${\mathcal O}_i$ be the ring of integers of the completion of $K$ at ${\mathfrak p}_i$. Let ${\mathcal O}_p={\mathcal O}_1^{k_1} \times \cdots \times {\mathcal O}_r^{k_r}$. Is there a description of subrings $R$ of ${\mathcal O}_p$ such that $[{\mathcal O}_p:R ] < \infty$ (additive subgroup index) ? Is it for example true that "most" of such $R$'s are of the form $R_1 \times \cdots \times R_r$? Here "most" should mean the following: Let ${\mathcal n}_t$ be the number of subrings $R$ such that $[{\mathcal O}_p: R] \leq t$ and let ${\mathcal n}'_t$ be the number of subrings $R$ of index at most $t$ which are expressible as a direct product $R_1 \times \cdots \times R_r$. Then is it true that $n'_t / n_t \to 1$ as $t \to \infty$? It would of course be most desirable if $n_t = n'_t$. My sincerest apologies ahead of time if this turns out to be a stupid question.

Added in revision: Thank you for the answer and the comment. I would like to think of Laurent's example as ${\mathbb Z}_p^2$ and not ${\mathbb Z}_p \times {\mathbb Z}_p$. So it seems that I posed the question incorrectly; what I really need is to group together those ${\mathcal O}_i$'s that are isomorphic to each other, even if the the corresponding ${\mathfrak p}_i$'s are different.

A word about the genesis of this problem: Nathan Kaplan and I have recently proved a theorem about counting the number of subrings of ${\mathbb Z}^n$ for small $n$ using $p$-adic integration techniques (we treat $n \leq 5$, the case $n=5$ seems to be new). Now I'm trying to see if we can use what we have proved to count orders in quintic fields (for cubic fields this is due to Davenport and Heilbronn, and for quartic fields this is a consequence of Nakagawa's work). To illustrate the idea, for a moment imagine that ${\mathbb Q}$ has a quintic Galois extension $K$. Except for finitely many exceptions, a given prime of $p$ either remains prime or splits as a product ${\mathfrak p}_1 \cdots {\mathfrak p}_5$. The situation for split primes is the same as counting the number of subrings of ${\mathbb Z}_p^5$ which we know how to do. The inert primes require separate treatment.

It turns out that the question is quite a bit more interesting than I had originally thought (what else is new!).