The following setup is standard studied in Algebraic Number Theory:
Let $K$ be a number field and $\mathcal{O}_K$ be its ring of integers. Suppose $[K : \mathbb{Q}] = n$ so we have $n$ embeddings of $K$ into $\mathbb{C}$. Suppose there are $r$ real embeddings and $2s$ complex embeddings (nonreal), then we have
$$ K \hookrightarrow \mathbb{R}^r \times \mathbb{C}^{2s} \cong \mathbb{R}^n$$ by mapping pairs of complex embeddings to the real and imaginary parts. Then we know that $\mathcal{O}_K$ maps into a (full-rank) lattice in $\mathbb{R}^n$ (that is, $\mathbb{Z}$ combinations of an $\mathbb{R}$ basis for $\mathbb{R}^n$). Similarly, any fractional ideal also maps into a lattice in $\mathbb{R}^n$.
I want to understand the following question:
What are the algebraic structures within $K$ that map to lattices in $\mathbb{R}^n$?
I understand that any free abelian group (free $\mathbb{Z}$-module) would be necessary but insufficient (because lattices in $\mathbb{R}^n$ has this discrete property, where accumulation points are disallowed). A finitely generated nonzero $\mathcal{O}_K$-module in $K$ (aka fractional ideals) would be sufficient but probably not necessary.
If there are any references that would be greatly appreciated!