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

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 $\mathbb{Z}$.

• Just a comment on your "natural definition"; how are you defining "not divisible by" in a field? The existence of inverses means that everything non-zero divides everything else, in the sense that for all $a,b\in K$ with $b\ne0$, there exists $c\in K$ (i.e. $c=a/b$) such that $a=bc$.
– mdp
Sep 8, 2014 at 12:38
• @MattPressland You're right. But then, I was thinking inside the ring, not field here. But then how I would be able to define this would be an issue though.
– FAM
Sep 8, 2014 at 12:39
• I see - but yes, then you have a logical loop.
– mdp
Sep 8, 2014 at 13:19
• This boils down to motivating the definition of an algebraic integer. Sep 8, 2014 at 13:43

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).