integral closure

Let $B$ be a ring with a subring $A$ . The integral closure of $A$ in $B$ is the set $A' \subset B$ consisting of all elements of $B$ which are integral over $A$ .

It is a theorem that the integral closure of $A$ in $B$ is itself a ring. In the special case where $A = \mathbb{Z}$ , the integral closure $A'$ of $\mathbb{Z}$ is often called the ring of integers in $B$ .