Subrings of the product of rings of algebraic integers

Question

Let $K,L$ be a number fields and $\mathcal{O}_K,\mathcal{O}_L$ their rings of algebraic integers, and $n_K=[K:\mathbb Q]$, $n_L=[L:\mathbb Q]$.

Suppose that $R$ is a subring of $\mathcal{O}_K\oplus\mathcal{O}_L$ such that $\text{rank}((R,+))=n_K+n_L$.

Is true or false that $R=\mathcal{O}_1\oplus\mathcal{O}_2$ where $\mathcal{O}_1,\mathcal{O}_2$ are subrings respectively of $\mathcal{O}_K,\mathcal{O}_L$ such that, the rank of $\mathcal{O}_1,\mathcal{O}_2$ are respectively $n_K,n_L$?

Thanks you all.

Answer 1

This is already false for $\mathbb{Z}$. There's a subring of $\mathbb{Z} \times \mathbb{Z}$ consisting of pairs of integers $(a, b)$ such that $a \equiv b \bmod 2$, which is of the correct rank but not of the form you want.