Why is ring of integers $\mathcal O_K$ called ring of integers - what properties of $\mathbb{Z}$ does it inherit?

Question

I was wondering why ring of integers $\mathcal O_K$ for field $K$ is called ring of integers.

Definition says that elements in this ring will be a solution for monic equation with coefficients of rational integers. But wouldn't natural definition of ring of integers be ring of elements cannot be expressed as $\frac{a}{b}$ where $a$ is not divisible by $b$?

So what properties of $\mathbb{Z}$ does this ring inherit?

Edit: One can just answer about what properties get inherited from $\mathbb{Z}$.

Answer 1

Just like $\mathbb Z$ is integrally closed (due to rational root theorem), one can show that $\mathcal O _K$ is also integrally closed, that is, if $\beta \in K$ is the root of a monic polynomial $f$ with coefficients in $\mathcal O _K$, then $\beta \in \mathcal O _K$. And they both are finitely-generated $\mathbb Z$-modules (have integral basis).