Statements (37)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
Module theory concept
|
| gptkbp:characterization |
A module is projective if and only if the functor Hom(P,-) is exact
|
| gptkbp:citation |
Atiyah, M. F.; Macdonald, I. G. Introduction to Commutative Algebra
Lang, Serge. Algebra Rotman, Joseph J. An Introduction to Homological Algebra |
| gptkbp:contrastsWith |
gptkb:Injective_module
Flat module |
| gptkbp:definedIn |
gptkb:King
|
| gptkbp:defines |
A module P such that every surjective module homomorphism f:M→N and every module homomorphism g:P→N, there exists a module homomorphism h:P→M such that f∘h=g.
|
| gptkbp:example |
gptkb:Finitely_generated_projective_module
gptkb:Free_module Direct sum of free modules |
| gptkbp:field |
gptkb:Abstract_algebra
gptkb:Ring_theory |
| gptkbp:generalizes |
gptkb:geometry
Projective representation |
| gptkbp:hasProperty |
Lifting property
Splitting property |
| https://www.w3.org/2000/01/rdf-schema#label |
Projective module
|
| gptkbp:namedFor |
Algebraists
|
| gptkbp:property |
Direct summand of a free module
Every free module is projective Every projective module is flat Projective modules are closed under direct sums Projective modules are closed under direct summands |
| gptkbp:relatedConcept |
Projective resolution
Projective cover Projective dimension |
| gptkbp:used_in |
gptkb:Algebraic_topology
gptkb:Algebraic_geometry gptkb:Representation_theory Category theory Homological algebra |
| gptkbp:bfsParent |
gptkb:Finitely_generated_projective_module
gptkb:Free_module gptkb:Stably_free_module |
| gptkbp:bfsLayer |
7
|