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.
Read more about this topic: Integrally Closed
Famous quotes containing the word rings:
“If a man do not erect in this age his own tomb ere he dies, he shall live no longer in monument than the bell rings and the widow weeps.”
—William Shakespeare (1564–1616)