Integrally Closed - Commutative Rings

Commutative Rings

A commutative ring contained in a ring is said to be integrally closed in if is equal to the integral closure of in . That is, for every monic polynomial f with coefficients in, every root of f belonging to S also belongs to . Typically if one refers to a domain being integrally closed without reference to an overring, it is meant that the ring is integrally closed in its field of fractions.

If the ring is not a domain, typically being integrally closed means that every local ring is an integrally closed domain.

Sometimes a domain that is integrally closed is called "normal" if it is integrally closed and being thought of as a variety. In this respect, the normalization of a variety (or scheme) is simply the of the integral closure of all of the rings.