Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Integral ring maps and integral extensions
Definition
Let be a homomorphism of commutative rings. The map is an integral ring map when every element of is integral over in the sense of Integral elements over a commutative ring and algebraic integers. When is identified with a subring of , one also says that is an integral extension of and writes integral.
Depends on
Used by
- Integral closure in an extension ring and integrally closed domains Definition
- For an integral extension of domains, the upper ring is a field if and only if the lower ring is Lemma
- Comparable primes with the same contraction are equal under an integral map Theorem
- Going down holds for integral extensions over integrally closed domains Theorem
- Going up for integral ring maps Theorem
- Integral extensions are transitive Theorem
- Integrality and integral closure commute with localisation Theorem
- Lying over for integral ring maps Theorem
Dependency tree · two levels
4 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 (10.21) (standard reference, not scraped)
- J. S. Milne, A Primer of Commutative Algebra, v4.03, Definition 6.6 (standard reference, not scraped)