Integrally closed domains also play a role in the hypothesis of the Going-down theorem.
2.
It is a standard algebra exercise to show this implies that " R " is an integrally closed domain.
3.
This implies in particular that an integral element over an integrally closed domain " A " has a minimal polynomial over " A ".
4.
If the homogeneous coordinate ring " R " is an integrally closed domain, then the projective variety " X " is said to be projectively normal.
5.
The restriction of algebraic field extensions to subrings has led to the notions of integral extensions and integrally closed domains as well as the notion of ramification of an extension of valuation rings.
6.
This is significant since the analog is false for an integrally closed domain : let " R " be a valuation domain of height at least 2 ( which is integrally closed . ) Then RX is not integrally closed.
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