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 principal ideal domain is Noetherian
Statement
Every principal ideal domain is Noetherian.
Facts & Assumptions
Given: A principal ideal domain , regarded as its left regular module, and the Noetherian-ring convention of Left and right Noetherian rings.
An integral domain is a principal ideal domain when every ideal is principal: for some (Principal ideal domain).
A module is Noetherian exactly when every submodule is finitely generated (Finite generation, ACC, and maximal-condition characterizations of Noetherian modules).
Proof
A submodule of the left regular module is an ideal. By [F1] it is generated by one element; this includes the zero ideal and unit ideal .
Thus every submodule of is finitely generated, so [L1] makes the regular module Noetherian and hence makes a Noetherian ring. Commutativity makes the same statement valid on the right.
Depends on
Used by
Dependency tree · two levels
7 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
- K. Conrad, Modules over a PID, Section 2 (standard reference, not scraped)