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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, by keeping some of the main properties of free modules. Various equivalent characterizations of these modules appear below. Projective modules were first introduced in 1956 in the influential book Homological Algebra by Henri Cartan and Samuel Eilenberg.

Attributes	Values
rdf:type	Thing
rdfs:label	Projektive Dimension (de) Módulo proyectivo (es) Module projectif (fr) Modulo proiettivo (it) 射影加群 (ja) 사영 가군 (ko) Projective module (en) Проективный модуль (ru) Проєктивний модуль (uk) 投射模 (zh)
rdfs:comment	Die projektive Dimension ist ein homologischer Begriff aus der kommutativen Algebra. Sie misst, wie weit ein Modul davon entfernt ist, projektiv zu sein. Ein projektiver Modul hat die projektive Dimension Null. Dieser Artikel beschäftigt sich mit kommutativer Algebra. Insbesondere sind alle betrachteten Ringe kommutativ und haben ein Einselement. Ringhomomorphismen bilden Einselemente auf Einselemente ab. Für weitere Details siehe Kommutative Algebra. (de) En mathématiques, un module projectif est un module P (à gauche par exemple) sur un anneau A tel que pour tout morphisme surjectif f : N → M entre deux A-modules (à gauche) et pour tout morphisme g : P → M, il existe un morphisme h : P → N tel que g = fh, c'est-à-dire tel que le diagramme suivant commute : Autrement dit : P est projectif si pour tout module N, tout morphisme de P vers un quotient de N se factorise par N. (fr) En matemáticas, particularmente en álgebra abstracta y álgebra homológica, el concepto de módulo proyectivo sobre un anillo R es una generalización más flexible de la idea de un módulo libre (es decir, un módulo con ). Hay varias caracterizaciones equivalentes de estos módulos. (es) 환론에서 사영 가군(射影加群, 영어: projective module)은 자유 가군을 직합으로 분해하였을 때의 한 성분으로 나타낼 수 있는 가군이다. 가군의 범주에서의 사영 대상이다. (ko) In matematica, un modulo proiettivo è un modulo con la proprietà di essere addendo diretto di un modulo libero: ovvero P è proiettivo se esiste un modulo libero F e un suo sottomodulo N tale che F è la somma diretta di P ed N. Questo concetto è il duale di quello di modulo iniettivo; è stato introdotto da Henri Cartan e Samuel Eilenberg nel 1956. (it) 数学において、射影加群(しゃえいかぐん、英: projective module）とは、表現可能関手 Hom(P, –) が完全となるような加群 P のことである。自由加群の一般化に相当する。ホモロジー代数学における基本的な概念のひとつであり、で導入された。 (ja) Проекти́вный мо́дуль — одно из основных понятий гомологической алгебры. С точки зрения теории категорий, проективные модули являются частным случаем проективных объектов. (ru) Проєктивний модуль — важливий тип модулів, що є узагальненням вільних модулів. З точки зору теорії категорій, проєктивні модулі є окремим випадком проєктивних об'єктів. (uk) 在交換代數中，一個環上的投射模是自由模的推廣，它有多種等價的定義；就幾何的觀點，投射模之於自由模一如向量叢之於平凡向量叢。在範疇論的語言中，投射模可以推廣為一個阿貝爾範疇中的投射對象。投射模首見於昂利·嘉當與塞繆爾·艾倫伯格的重要著作 Homological Algebra，由此定義的投射分解是同調代數的基本概念之一。 (zh) 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, by keeping some of the main properties of free modules. Various equivalent characterizations of these modules appear below. Projective modules were first introduced in 1956 in the influential book Homological Algebra by Henri Cartan and Samuel Eilenberg. (en)
foaf:depiction
dcterms:subject	Module theory Homological algebra
Wikipage page ID	364488 (xsd:integer)
Wikipage revision ID	1119863732 (xsd:integer)
Link from a Wikipage to another Wikipage	Prime ideal Principal ideal Principal ideal domain Samuel Eilenberg Epimorphism Module (mathematics) Dedekind ring Algebraic K-theory Module theory Paulo Ribenboim Vector space Dedekind domain Injective module Integral domain Lifting property Projective cover Compact space Countable set Mathematical induction Mathematics Noetherian ring Subring Quotient module Class (set theory) Converse (logic) Left exact functor Localization (commutative algebra) Short exact sequence Commutative ring Zero ring Functor Deep result Ideal (ring theory) Idempotent (ring theory) Koszul complex Quillen–Suslin theorem Spectrum of a ring Matrix ring Maximal ideal Domain (ring theory) Dual (category theory) Hausdorff space Local ring Algebra Algebra over a ring Field (mathematics) Finitely generated module Flat module Nicolas Bourbaki Charles Weibel Dimension (vector space) Direct limit Direct sum of modules Faithfully flat descent Isomorphism Kaplansky's theorem on projective modules Faithfully flat morphism Finitely presented module Mathematical proof Projection (linear algebra) Projective object Resolution (algebra) Ring (mathematics) Group (mathematics) Henri Cartan Addison–Wesley Counterexample Surjective Homological algebra Abelian group Hereditary ring Torsion-free module Modular representation theory

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