|
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 and are discussed in some detail in the textbook . 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. == Definition == A left module ''Q'' over the ring ''R'' is injective if it satisfies one (and therefore all) of the following equivalent conditions: * If ''Q'' is a submodule of some other left ''R''-module ''M'', then there exists another submodule ''K'' of ''M'' such that ''M'' is the internal direct sum of ''Q'' and ''K'', i.e. ''Q'' + ''K'' = ''M'' and ''Q'' ∩ ''K'' = . * Any short exact sequence 0 →''Q'' → ''M'' → ''K'' → 0 of left ''R''-modules splits. * If ''X'' and ''Y'' are left ''R''-modules and ''f'' : ''X'' → ''Y'' is an injective module homomorphism and ''g'' : ''X'' → ''Q'' is an arbitrary module homomorphism, then there exists a module homomorphism ''h'' : ''Y'' → ''Q'' such that ''hf'' = ''g'', i.e. such that the following diagram commutes: :: * The contravariant functor Hom(-,''Q'') from the category of left ''R''-modules to the category of abelian groups is exact. Injective right ''R''-modules are defined in complete analogy. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「injective module」の詳細全文を読む スポンサード リンク
|