Normal domain

Definition
An integral domain $$R$$ is said to be normal if it satisfies the following equivalent conditions:


 * 1) $$R$$ is integrally closed in its field of fractions.
 * 2) If $$S$$ is a subring of the field of fractions of $$R$$ that contains $$R$$ as a subgroup of finite index, then $$S = R$$.

Metaproperties
In fact, the localization at a multiplicatively closed subset of a normal domain continues to be a normal domain.

The polynomial ring in one variable over a normal domain is again normal.

Ring of integer-valued polynomials
The ring of integer-valued polynomials over a normal domain is again a normal domain.