Alphabeta MathHow statement and proof provenance work
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Integral closure in an extension ring and integrally closed domains
Definition
Let be a domain and let be a homomorphism into a commutative ring. The integral closure of in is the set of elements of integral over . When is a field extension of the field of fractions of The field of fractions of an integral domain, the integral closure of in is often denoted .
The domain is integrally closed when every element of integral over already lies in . Thus an integrally closed domain is one whose field of fractions contains no new elements integral over it.
Depends on
Used by
- Minimal polynomials of integral elements over an integrally closed domain have coefficients in the domain Lemma
- A domain is integrally closed if and only if its prime localisations are, equivalently if and only if its maximal localisations are Theorem
- Integrality and integral closure commute with localisation Theorem
- The integral closure of a domain in a field extension is integrally closed Theorem
Dependency tree · two levels
8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Allen B. Altman and Steven L. Kleiman, A Term of Commutative Algebra, 13th ed., Definition (10.30) (standard reference, not scraped)
- J. S. Milne, A Primer of Commutative Algebra, v4.03, Definitions 6.6 and 6.9 (standard reference, not scraped)