Research any topic before you write.

Find related topics. | Discover entities. | See connections. | Build a topical map.

Projective module

In mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules (that is, modules with basis vectors) over a ring, keeping some of the main properties of free modules. Various equivalent characterizations of these modules appear below.

Art, Definitions & Relation to other module-theoretic properties

Use the mouse wheel or two fingers (on touchscreens) to zoom in and out of the map.

Research this topic

Explore the main themes, entities and connections around Projective module. Start with the topic map, then use the sections below for research and deeper semantic analysis.

Explore this topic

Start with a few of the strongest sections from the source topic. These are research directions, not a list of keywords you must use.

Definitions

Category theoreticalLiftingIf and only ifSurjectiveModule homomorphism
25 related topics

Relation to other module-theoretic properties

FlatTorsion-free modulesDomainLocal ringSkew field
28 related topics

Projective modules over commutative rings

LocalizationPrime idealNoetherian ringsBoolean ringFinitely presented modules
17 related topics

Vector bundles and locally free modules

Vector bundlesContinuousRealCompactHausdorff space
10 related topics

Topics to explore

Browse the full topic structure. Each item opens a new analysis centered on that subject.

Overview

Definitions

Elementary examples and properties

Relation to other module-theoretic properties

The category of projective modules

Projective resolutions

Projective modules over commutative rings

Vector bundles and locally free modules

Projective modules over a polynomial ring

Advanced semantic analysis

Deeper signals for content research, entity SEO and topical coverage. The plain-language headings explain what each technical view is useful for.

Map overview Semantic statistics

Projective module

Nodes124
Edges123
Triples162
Avg. degree1.98
Density0.016129
Components1

How this topic connects Entity context

See the strongest relationship patterns around the current topic before diving into the raw triples.

Projective module

Top relations

related to References · 96
Projective module → Adamson, Addison, Adkins, Algebra, Algebras, AMS Chelsea, An, An Approach, Ann, Autour, BF01390094, Bibcode, Birkhäuser Boston, Boyd, Bulletin, Ch, Charles Weibel, Cohn, Commutative, Course
related to Projective modules over a polynomial ring · 13
Projective module → Bass, Quillen, R-module, Serre, Serre's, Since, Suslin, The, The Quillen, This, Thus, X1, Xn
related to Projective vs. free modules · 7
Projective module → Any, For, In, Kaplansky, Kaplansky's, The, This
related to Projective resolutions · 6
Projective module → Every, Given, In, Koszul, Pi, The
related to Relation to other module-theoretic properties · 6
Projective module → For, PID, Quillen, Suslin, The, There
related to Vector bundles and locally free modules · 6
Projective module → Hausdorff, If, Serre, Swan, This, Vector
related to Lifting property · 5
Projective module → It, R-modules, The, Thus, We
related to Elementary examples and properties · 4
Projective module → Direct, If, Re, The
related to Projective modules over commutative rings · 4
Projective module → Noetherian, Projective, The, Thus
related to Projective vs. flat modules · 4
Projective module → Conversely, Every, The, Z-module

Important terminology Word statistics

Use these terms to understand the vocabulary surrounding the topic, not as a checklist for keyword stuffing.

Important terminology

projective free modules module ring ideal every rings displaystyle flat commutative finitely generated principal direct locally domain isbn field algebra

Entity relationships Subject–Predicate–Object triples

SubjectPredicateObjectConfidenceSrc
Projective moduleis afree module if the ring is a principal ideal domain such as the integers0.90text
Projective moduleis aprojective module over the localized ring0.90text
the integersinstance ofevery projective module is a free module if the ring is a principal ideal domain0.80text
or ainstance ofevery projective module is a free module if the ring is a principal ideal domain0.80text
Projective modulerelated to Elementary examples and propertiesThe0.60section
Projective modulerelated to Elementary examples and propertiesDirect0.60section
Projective modulerelated to Elementary examples and propertiesIf0.60section
Projective modulerelated to Elementary examples and propertiesRe0.60section
Projective modulerelated to Lifting propertyThe0.60section
Projective modulerelated to Lifting propertyWe0.60section
Projective modulerelated to Lifting propertyIt0.60section
Projective modulerelated to Lifting propertyThus0.60section

Related concept clusters Concept neighborhoods

These clusters group vocabulary that occurs around closely connected concepts in the source material.

    Connections between topic areas Semantic bridges

    Bridges can reveal useful research angles that are easy to miss in a flat list of related terms.

    Min side: 3
    For writers, content strategists, SEOs, marketers and creators — from quick topic research to advanced semantic analysis.