DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-31

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.

Prime-ideal valuations on fractional ideals

Definition

Let $R$ be a Dedekind domain, let $p$ be a nonzero prime ideal, and let $I$ be a nonzero fractional ideal of $R$ . By Localizing a Dedekind domain at a nonzero prime gives a DVR, the localisation $R_{p}$ is a DVR, and by Every nonzero fractional ideal of a Dedekind domain is invertible together with Ideals in a DVR are powers of the maximal ideal the localised ideal has the form

I_{p} = p^{n} R_{p}

for a unique integer $n$ . This integer is the prime-ideal valuation of $I$ at $p$ , written $v_{p} (I) : = n$ .

Depends on

Used by

Dependency tree · two levels

12 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

J. P. May, Notes on Dedekind Rings (standard reference, not scraped)
J. S. Milne, A Primer of Commutative Algebra, §20 (standard reference, not scraped)