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, 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.
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...
algebraically compact modules are analogous to injectivemodules, where one can extend all module homomorphisms. All injectivemodules are algebraically compact...
particularly in algebra, the injective hull (or injective envelope) of a module is both the smallest injectivemodule containing it and the largest essential...
field of category theory, the concept of injective object is a generalization of the concept of injectivemodule. This concept is important in cohomology...
generated by injectivemodules is injective. The converse is a result of (Matlis 1958): if every module has a unique maximal injective submodule, then...
if every direct sum of injective (left/right) modules is injective. Every left injectivemodule over a left Noetherian module can be decomposed as a direct...
injective hull) is a maximal essential extension, or a minimal embedding in an injectivemodule. 3. An injective cogenerator is an injectivemodule such...
Differential module Five lemma Short five lemma Snake lemma Nine lemma Extension (algebra) Central extension Splitting lemma Projective moduleInjectivemodule Projective...
necessarily an injectivemodule, and is unique up to isomorphism. The injective hull is also minimal in the sense that any other injectivemodule containing...
_{R}S} is injective. Hence, M → M ⊗ R S {\displaystyle M\to M\otimes _{R}S} is injective. Conversely, if M ≠ 0 {\displaystyle M\neq 0} is a module over R...
more general than module categories: we don't need a notion of "free object". It can also be dualized, leading to injectivemodules. The lifting property...
{\displaystyle s\in S} to itself) is injective. In particular, the identity function X → X {\displaystyle X\to X} is always injective (and in fact bijective). If...
{\displaystyle \Omega ^{-1}} can be defined as follows. Given M, find an injectivemodule I with an inclusion i : M → I {\displaystyle i\colon M\to I} . Then...
algebra, a module homomorphism is a function between modules that preserves the module structures. Explicitly, if M and N are left modules over a ring...
In some loose sense, it is an analog of the defining diagram of an injectivemodule in abstract algebra. The most basic example is a CW pair ( X , A )...
thus a faithful contravariant functor from left R-modules to right R-modules. Every H* is pure-injective (also called algebraically compact). One can often...
coprimary modules. For a one-sided Noetherian ring, there is a surjection from the set of isomorphism classes of indecomposable injectivemodules onto the...
German mathematician, known for his work in algebra. He introduced injectivemodules in 1940. He is the eponym of Baer rings and Baer groups. Baer studied...
envelope or injective hull of a module is a smallest injectivemodule containing it. 3. An injective resolution is a resolution by injectivemodules. 4. The...
discovered by Joachim Lambek shows that a module is flat if and only if the associated character module is injective. The group ( Q / Z , + ) {\displaystyle...
If the module is an injectivemodule, then indecomposability is equivalent to the endomorphism ring being a local ring. For a semisimple module, the endomorphism...
example the Leray spectral sequence. An injective sheaf F {\displaystyle {\mathcal {F}}} is a sheaf that is an injective object of the category of abelian sheaves;...
Bass (1963, p.11). The Bass numbers describe the minimal injective resolution of a finitely-generated module M over a Noetherian ring: for each prime ideal p...
homomorphism is an isomorphism. A torsionless module is one for which the canonical homomorphism is injective. Example: If G = Spec ( A ) {\displaystyle...
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