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Canonical static concept record

Residue field

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Summary

For a local ring $(R,\mathfrak m)$ , the quotient field $R/\mathfrak m$ .

Record metadata

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

Carrier(s)

local ring $(R,\mathfrak m)$

Data

quotient $R/\mathfrak m$

Canonically induces

Reduction modulo the maximal ideal

Concept sources

Wikipedia — Residue field — Residue field · encyclopedic reference · source ID wp-residue-field
nLab — Residue field — Residue field · mathematical reference · source ID nlab-residue-field
Stacks Project — Commutative Algebra (PDF) — Commutative Algebra (PDF) · reference text · source ID stacks-algebra-text

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

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

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.

Residue field Field

Permalink to relation

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

Relation ID: story_residue_field_to_field_induced_forgotten
Relation type: Induced / forgotten induced-forgotten
Direction: source → target
Endpoint roles: source: Yields by induction / forgetting; target: Obtained by induction / forgetting from
Authored annotation: is a field

Authored explanation

The quotient of a local ring by its unique maximal ideal is a field.

How to interpret this relation type

Pass canonically from a stronger object to structure it determines, or forget part of the data while retaining a valid weaker structure. The carrier may change under a canonical induced construction.

Relation sources

Stacks Project — Commutative Algebra (PDF) — Commutative Algebra (PDF) · reference text · source ID stacks-algebra-text
Wikipedia — Field (mathematics) — Field (mathematics) · expository · source ID wp-field
nLab — Field — Field · reference · source ID nlab-field

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Residue field

Summary

Record metadata

Carrier(s)

Data

Canonically induces

Concept sources

Incoming relations (arrows to this concept)

Local ring Residue field

Authored explanation

How to interpret this relation type

Relation sources

Outgoing relations (arrows from this concept)

Residue field Field

Authored explanation

How to interpret this relation type

Relation sources