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Canonical static concept record

Injective module

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Summary

A module $I$ for which $\operatorname{Hom}(-,I)$ is exact, equivalently maps into $I$ extend across injective module maps.

Record metadata

Canonical name: Injective module
Concept ID: injective_module
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Homological algebra
Domains: Homological algebra, Algebra
Alternate terminology: No alternate terminology is currently authored for this record.

Carrier(s)

ring $R$
$R$ -module $I$

Axioms / constraints

extension property against injective $R$ -module homomorphisms

Canonically induces

injective resolutions because module categories have enough injectives

Notes

Injectivity depends on the base ring and is categorical, not the same as injectivity of a function.

Concept sources

Wikipedia — Injective module — Injective module · encyclopedic reference · source ID wp-injective-module
nLab — injective module — injective module · mathematical reference · source ID nlab-injective-module
Stacks Project — Injective modules — Injective modules · mathematical reference · source ID stacks-injective-module

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Module over a ring Injective module

Permalink to relation

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

Relation ID: module_to_injective_module
Relation type: Impose axiom impose-axiom
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: + injective extension property

Authored explanation

Require maps into the module to extend across every injective module homomorphism, equivalently require $\operatorname{Hom}(-,I)$ to be exact.

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 — Injective module — Injective module · encyclopedic reference · source ID wp-injective-module
nLab — injective module — injective module · mathematical reference · source ID nlab-injective-module
Stacks Project — Injective modules — Injective modules · mathematical reference · source ID stacks-injective-module

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.

No direct relations are authored in this direction.

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