integral closure

Let $B$ be a ring with a subring $A$. The integral closure of $A$ in $B$ is the set $A^{\prime }\subset B$ 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 $A^{\prime }$ of $\mathbb{Z}$ is often called the ring of integers in $B$.