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.
Every DVR is a PID
Statement
Every discrete valuation ring is a principal ideal domain.
Facts & Assumptions
Given: A discrete valuation ring .
Every nonzero ideal of a DVR is a power of its maximal ideal, hence principal (Ideals in a DVR are powers of the maximal ideal).
A principal ideal domain is an integral domain in which every ideal is principal (Principal ideal domain).
Proof
A discrete valuation ring is a domain. Its zero ideal is principal, and [L1] shows that every nonzero ideal is principal.
Therefore every ideal of the domain is principal, so [F1] makes a principal ideal domain.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
6 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. S. Milne, A Primer of Commutative Algebra, v4.03, Proposition 20.2 (standard reference, not scraped)
- Allen B. Altman and Steven L. Kleiman, A Term of Commutative Algebra, 13th ed., (23.1) (standard reference, not scraped)