Ring of integers in a number field

Definition
Let $$K$$ be a number field. Then the ring of integers in $$K$$, denoted $$O_K$$, is the subring of $$K$$ comprising all those elements which are integral over $$\mathbb{Z}$$, in other words, which satisfy monic polynomials with integer coefficients.

The ring of integers in a number field is also termed a maximal order, here an order is a subring of $$K$$ which is free as a $$\mathbb{Z}$$-module, and which, over $$\mathbb{Q}$$, generates $$K$$.

Weaker properties

 * Dedekind domain