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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

Module over a ring

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Summary

An abelian group acted on linearly by a ring.

Record metadata

Canonical name: Module over a ring
Concept ID: module
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Algebra
Domains: Algebra, Representation theory, Homological algebra
Alternate terminology: No alternate terminology is currently authored for this record.

Carrier(s)

scalar ring $R$
carrier set $M$

Data

abelian-group addition on $M$
scalar multiplication $R \times M \to M$

Axioms / constraints

module axioms

Notes

Convention: modules are left modules unless explicitly stated otherwise. Right modules reverse the scalar-action convention.

Concept sources

Wikipedia — Module (mathematics) — Module (mathematics) · expository · source ID wp-module
nLab — Module — Module · reference · source ID nlab-module
Stacks Project — Modules — Modules · reference · source ID stacks-modules
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.

ring action on an abelian group(construction junction)Module over a ring

Permalink to relation

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

Relation ID: e_jmodule_module
Relation type: Combine compatible structures combine-compatible
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: + scalar action

Authored explanation

Add compatible scalar multiplication.

How to interpret this relation type

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

Relation sources

Wikipedia — Module (mathematics) — Module (mathematics) · expository · source ID wp-module
nLab — Module — Module · reference · source ID nlab-module
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.