If a ring has its field of fraction as algebraic number field $K$, would this ring be $O_K$?

Question

Suppose that ring has its field of fraction as algebraic number field $K$. Would this ring then be $O_K$, ring of integers?

Also, for $O_K$, would subring of $O_K$ be integrally closed?

Answer 1

The first assertion is not true. For example consider the ring $O=\mathbb{Z}[\sqrt{-3}]$ in $K=\mathbb{Q}(\sqrt{-3})$. In this case $O_K = \mathbb{Z}[\frac{1+\sqrt{-3}}{2}]$, and $O \subsetneq O_K$, but $Frac(O)= K$.

The meaning of your second question is not so clear. Are you asking if a subring of $O_K$ is integrally closed in $K$ over $\mathbb{Z}$? The ring $O_K$ is by definition the integral closure of $\mathbb{Z}$ in $K$, so $O_K$ itself is integrally closed (in $K$ over $\mathbb{Z}$), and any proper subring of $O_K$ is not.