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 injective modules, where one can extend all module homomorphisms. All injective modules are algebraically compact...

injective hull) is a maximal essential extension, or a minimal embedding in an injective module. 3.  An injective cogenerator is an injective module such...

generated by injective modules is injective. The converse is a result of (Matlis 1958): if every module has a unique maximal injective submodule, then...

field of category theory, the concept of injective object is a generalization of the concept of injective module. This concept is important in cohomology...

if every direct sum of injective (left/right) modules is injective. Every left injective module over a left Noetherian module can be decomposed as a direct...

particularly in algebra, the injective hull (or injective envelope) of a module is both the smallest injective module containing it and the largest essential...

projective modules or flat modules. Similarly every module has injective resolutions, which are right resolutions consisting of injective modules. Given a...

many of their desirable properties. Injective Injective modules are defined dually to projective modules. Flat A module is called flat if taking the tensor...

more general than module categories: we don't need a notion of "free object". It can also be dualized, leading to injective modules. The lifting property...

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...

g\circ f} is injective, then it can only be concluded that f {\displaystyle f} is injective (see figure). Every embedding is injective. A function is...

This implies that, if f : M → N {\displaystyle f\colon M\to N} is an injective module homomorphism, then S − 1 R ⊗ R f : S − 1 R ⊗ R M → S − 1 R ⊗ R N {\displaystyle...

Differential module Five lemma Short five lemma Snake lemma Nine lemma Extension (algebra) Central extension Splitting lemma Projective module Injective module Projective...

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;...

_{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...

homomorphism is an isomorphism. A torsionless module is one for which the canonical homomorphism is injective. Example: If G = Spec ⁡ ( A ) {\displaystyle...

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...

necessarily an injective module, and is unique up to isomorphism. The injective hull is also minimal in the sense that any other injective module containing...

Projective module Projective cover Swan's theorem Quillen–Suslin theorem Injective module Injective hull Flat module Flat cover Coherent module Finitely-generated...

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...

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...

{\displaystyle \Omega ^{-1}} can be defined as follows. Given M, find an injective module I with an inclusion i : M → I {\displaystyle i\colon M\to I} . Then...

finitely generated modules. For example, if f : M → M is a surjective R-endomorphism of a finitely generated module M, then f is also injective, and hence is...

ideal Hilbert's Nullstellensatz Flat module Flat map Flat map (ring theory) Projective module Injective module Cohen-Macaulay ring Gorenstein ring Complete...

If the module is an injective module, then indecomposability is equivalent to the endomorphism ring being a local ring. For a semisimple module, the endomorphism...

{\displaystyle B\cong f(A)\oplus s(C).} In particular, any module over a semisimple ring is injective and projective. Since "projective" implies "flat", a semisimple...

residue field k, and choose E to be an injective hull of k (sometimes called a Matlis module). The dual DR(M) of a module M is defined to be HomR(M,E). Then...

left modules over R is the category whose objects are all left modules over R and whose morphisms are all module homomorphisms between left R-modules. For...

German mathematician, known for his work in algebra. He introduced injective modules in 1940. He is the eponym of Baer rings, Baer groups, and Baer subplanes...