Injective module: Difference between revisions

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{{Short description|Mathematical object in abstract algebra}}

In [[mathematics]], especially in the area of [[abstract algebra]] known as [[module theory]], an '''injective module''' is a [[module (mathematics)|module]] ''Q'' that shares certain desirable properties with the '''Z'''-module '''Q''' of all [[rational number]]s. 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 (category theory)|dual]] to that of [[projective module]]s. Injective modules were introduced in {{harv|Baer|1940}} and are discussed in some detail in the textbook {{harv|Lam|1999|loc=§3}}.

Injective modules have been heavily studied, and a variety of additional notions are defined in terms of them: [[Injective cogenerator]]s 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 resolutions|injective dimension]] and represent modules in the [[derived category]]. [[Injective hull]]s are maximal [[essential extension]]s, and turn out to be minimal injective extensions. Over a [[Noetherian ring]], every injective module is uniquely a direct sum of [[indecomposable module|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 ring]]s of [[finite group]]s over [[field (mathematics)|field]]s. Injective modules include [[divisible group]]s and are generalized by the notion of [[injective object]]s in [[category theory]].