DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-13

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.

Principal localisation $R_{f} = {1, f, f^{2}, \dots}^{- 1} R$

Definition

For a commutative ring $R$ and $f \in R$ , the powers $S_{f} = {1, f, f^{2}, \dots} = {f^{n} : n \in N}$ form a multiplicative subset. The principal localisation of $R$ at $f$ is $R_{f} = S_{f}^{- 1} R .$ Its elements may be written $r / f^{n}$ . In particular, $R_{1}$ is canonically isomorphic to $R$ , while $R_{0}$ is the zero ring.

Depends on

Multiplicative subsets and the localisation $S^{-1}R$ as equivalence classes of fractions

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 14 results over 10 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

The Stacks Project, Section 10.9: Localization (standard reference, not scraped)

Principal localisation Rf={1,f,f2,…}−1RRf​={1,f,f2,…}−1R