DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-29

How 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 closure in an extension ring and integrally closed domains

Definition

Let $A$ be a domain and let $A \to B$ be a homomorphism into a commutative ring. The integral closure of $A$ in $B$ is the set of elements of $B$ integral over $A$ . When $K$ is a field extension of the field of fractions $Frac (A)$ of The field of fractions $Frac (D) = (D ∖ {0})^{- 1} D$ of an integral domain, the integral closure of $A$ in $K$ is often denoted $\overline{A}$ .

The domain $A$ is integrally closed when every element of $Frac (A)$ integral over $A$ already lies in $A$ . Thus an integrally closed domain is one whose field of fractions contains no new elements integral over it.

Integral closure in an extension ring and integrally closed domains

Definition

Depends on

Dependency tree · 2 levels

Sources