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- Injective_module abstract "In mathematics, especially in the area of abstract algebra known as module theory, an injective module is a module Q that shares certain desirable properties with the Z-module Q of all rational numbers. Specifically, if Q is a submodule of some other module, then it is already a direct summand of that module; also, given a submodule of a module Y, then any module homomorphism from this submodule to Q can be extended to a homomorphism from all of Y to Q. This concept is dual to that of projective modules. Injective modules were introduced in (Baer 1940) and are discussed in some detail in the textbook (Lam 1999, §3).Injective modules have been heavily studied, and a variety of additional notions are defined in terms of them: Injective cogenerators are injective modules that faithfully represent the entire category of modules. Injective resolutions measure how far from injective a module is in terms of the injective dimension and represent modules in the derived category. Injective hulls are maximal essential extensions, and turn out to be minimal injective extensions. Over a Noetherian ring, every injective module is uniquely a direct sum of indecomposable modules, and their structure is well understood. An injective module over one ring, may not be injective over another, but there are well-understood methods of changing rings which handle special cases. Rings which are themselves injective modules have a number of interesting properties and include rings such as group rings of finite groups over fields. Injective modules include divisible groups and are generalized by the notion of injective objects in category theory.".
- Injective_module thumbnail Injective_module.png?width=300.
- Injective_module wikiPageExternalLink ?id=PswhrD_wUIkC.
- Injective_module wikiPageExternalLink 1103039896.
- Injective_module wikiPageExternalLink p363.
- Injective_module wikiPageID "473775".
- Injective_module wikiPageRevisionID "599074636".
- Injective_module b "A".
- Injective_module hasPhotoCollection Injective_module.
- Injective_module p "N".
- Injective_module subject Category:Homological_algebra.
- Injective_module subject Category:Module_theory.
- Injective_module comment "In mathematics, especially in the area of abstract algebra known as module theory, an injective module is a module Q that shares certain desirable properties with the Z-module Q of all rational numbers. Specifically, if Q is a submodule of some other module, then it is already a direct summand of that module; also, given a submodule of a module Y, then any module homomorphism from this submodule to Q can be extended to a homomorphism from all of Y to Q.".
- Injective_module label "Injective module".
- Injective_module label "Module injectif".
- Injective_module label "Modulo iniettivo".
- Injective_module label "Módulo inyectivo".
- Injective_module label "Инъективный модуль".
- Injective_module label "內射模".
- Injective_module sameAs Αμφιμονοσήμαντο_πρότυπο.
- Injective_module sameAs Módulo_inyectivo.
- Injective_module sameAs Module_injectif.
- Injective_module sameAs Modulo_iniettivo.
- Injective_module sameAs m.02dtnj.
- Injective_module sameAs Q2716519.
- Injective_module sameAs Q2716519.
- Injective_module wasDerivedFrom Injective_module?oldid=599074636.
- Injective_module depiction Injective_module.png.
- Injective_module isPrimaryTopicOf Injective_module.
