Integral closure

Summary

The integral closure of a ring $A$ in a larger commutative ring $B$ is the set of elements of $B$ that are integral over $A$. It is a subring of $B$ that contains $A$, and it is the largest subring of $B$ that is integral over $A$. For an integral domain $A$, its integral closure in its own field of fractions is called its normalization. The ring of integers of a number field is the integral closure of $\Z$ in that field.