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Construction junctions are diamonds. They show where multiple structures must coexist on the same carrier and satisfy compatibility conditions.

Canonical static concept record

Local ring

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Summary

A commutative ring with exactly one maximal ideal.

Record metadata

Canonical name: Local ring
Concept ID: local_ring
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Algebraic geometry
Domains: Algebraic geometry, Algebra, Number theory
Alternate terminology: No alternate terminology is currently authored for this record.

Carrier(s)

commutative ring $R$

Axioms / constraints

unique maximal ideal

Canonically induces

residue field

Concept sources

Stacks Project — Local rings — Local rings · reference · source ID stacks-local-ring
Wikipedia — Local ring — Local ring · expository · source ID wp-local-ring
nLab — Local ring — Local ring · reference · source ID nlab-local-ring
Stacks Project — Schemes (PDF) — Schemes (PDF) · reference text · source ID stacks-schemes-text

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Commutative ring Local ring

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: e_comm_local
Relation type: Impose axiom impose-axiom
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: + unique maximal ideal

Authored explanation

Require exactly one maximal ideal.

How to interpret this relation type

Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.

Relation sources

Stacks Project — Local rings — Local rings · reference · source ID stacks-local-ring
Wikipedia — Local ring — Local ring · expository · source ID wp-local-ring
nLab — Local ring — Local ring · reference · source ID nlab-local-ring
Stacks Project — Schemes (PDF) — Schemes (PDF) · reference text · source ID stacks-schemes-text

Localization of a commutative ring Local ring

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This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: story_ring_localization_to_local_ring_framework_specialization
Relation type: Framework specialization framework-specialization
Direction: source → target
Endpoint roles: source: Specializes to; target: Special case or refinement of
Authored annotation: localize at a prime

Authored explanation

For a prime ideal $\mathfrak p$ , the localization $R_{\mathfrak p}$ is a local ring.

How to interpret this relation type

A more specific theory, model, entity class, or regime is obtained by restricting or extending the scope of a broader framework.

Relation sources

Stacks Project — Commutative Algebra (PDF) — Commutative Algebra (PDF) · reference text · source ID stacks-algebra-text

Valuation ring Local ring

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: e_valuation_local
Relation type: Theorem implication theorem-implication
Direction: source → target
Endpoint roles: source: Implies by theorem; target: Follows by theorem from
Authored annotation: canonically local

Authored explanation

Every valuation ring is a local domain.

How to interpret this relation type

Record a genuine theorem implication that is not part of the target definition; these edges may point toward a weaker structure.

Relation sources

Stacks Project — Valuation rings — Valuation rings · reference · source ID stacks-valuation-ring
Wikipedia — Valuation ring — Valuation ring · expository · source ID wp-valuation-ring
nLab — Valuation ring — Valuation ring · reference · source ID nlab-valuation-ring
Stacks Project — Local rings — Local rings · reference · source ID stacks-local-ring
Wikipedia — Local ring — Local ring · expository · source ID wp-local-ring
nLab — Local ring — Local ring · reference · source ID nlab-local-ring
Stacks Project — Schemes (PDF) — Schemes (PDF) · reference text · source ID stacks-schemes-text

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.

Local ring Complete local ring

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: e_local_complete
Relation type: Impose axiom impose-axiom
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: + $m$ -adic completeness

Authored explanation

Require the canonical map $R\to \varprojlim R/m^{n}$ to be an isomorphism.

How to interpret this relation type

Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.

Relation sources

Wikipedia — Complete ring — Complete ring · expository · source ID wp-complete-ring
nLab — completion of a ring — completion of a ring · mathematical reference · source ID nlab-completion-ring
Stacks Project — Schemes (PDF) — Schemes (PDF) · reference text · source ID stacks-schemes-text

Local ring Noetherian + local regularity(construction junction)

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This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: j_regular_local__local_ring
Relation type: Combine compatible structures combine-compatible
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: local structure

Authored explanation

Supply the Local ring structure as one jointly required input.

How to interpret this relation type

Feed several structures into a construction junction and impose compatibility between them.

Relation sources

Stacks Project — DVR and regular local ring equivalences — DVR and regular local ring equivalences · reference · source ID stacks-dvr-regular
Wikipedia — Regular local ring — Regular local ring · expository · source ID wp-regular-local-ring
Stacks Project — Schemes (PDF) — Schemes (PDF) · reference text · source ID stacks-schemes-text

Local ring Residue field

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: story_local_ring_to_residue_field_quotient_construction
Relation type: Quotient construction quotient-construction
Direction: source → target
Endpoint roles: source: Quotients to; target: Obtained as quotient of
Authored annotation: quotient by the maximal ideal

Authored explanation

For a local ring $(R,\mathfrak m)$ , take representatives $r\in R$ and identify $r\sim s$ exactly when $r-s\in\mathfrak m$ ; the quotient $R/\mathfrak m$ is the residue field.

How to interpret this relation type

Identify elements by a stated equivalence relation and equip the quotient with the induced structure. The edge detail must state the representatives and equivalence relation.

Relation sources

Stacks Project — Commutative Algebra (PDF) — Commutative Algebra (PDF) · reference text · source ID stacks-algebra-text

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Local ring

Summary

Record metadata

Carrier(s)

Axioms / constraints

Canonically induces

Concept sources

Incoming relations (arrows to this concept)

Commutative ring Local ring

Authored explanation

How to interpret this relation type

Relation sources

Localization of a commutative ring Local ring

Authored explanation

How to interpret this relation type

Relation sources

Valuation ring Local ring

Authored explanation

How to interpret this relation type

Relation sources

Outgoing relations (arrows from this concept)

Local ring Complete local ring

Authored explanation

How to interpret this relation type

Relation sources

Local ring Noetherian + local regularity(construction junction)

Authored explanation

How to interpret this relation type

Relation sources

Local ring Residue field

Authored explanation

How to interpret this relation type

Relation sources