An element of an -algebra is integral over when it satisfies a monic polynomial with coefficients in .
If is integral and is a maximal ideal of , then is maximal in . Indeed, is an integral domain integral over , and a subring over which a field is integral is itself a field.
For an integral extension and , localization gives a bijectionIt combines the prime ideal correspondence for localization with the contraction of a maximal ideal under an integral extension.
If are integral over , then is a finitely generated -module. In particular, every element of this subalgebra is integral over .
An integral domain is integrally closed when every element of its fraction field that is integral over the domain already belongs to the domain.
If a graded ring is a graded extension of a graded subring , then the integral closure of in is graded: every homogeneous component of an integral element is integral. This follows by separating the extremal degrees in a monic integral equation and inducting on the number of nonzero homogeneous components.
Let be an integral extension of domains with integrally closed. Given primes of and a prime of over , there is a prime over .
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An integral element is a term used in various fields, primarily in mathematics and abstract algebra, as well as in related contexts like computer science and physics. However, without a specific context, the meaning can vary. 1. **Mathematics/Abstract Algebra**: In ring theory, an integral element refers to an element of an integral domain (a type of commutative ring) that satisfies a monic polynomial equation with coefficients from that domain.