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...
28 KB (3,919 words) - 11:05, 18 May 2024
algebraically compact modules are analogous to injective modules, where one can extend all module homomorphisms. All injective modules are algebraically compact...
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particularly in algebra, the injective hull (or injective envelope) of a module is both the smallest injective module containing it and the largest essential...
8 KB (1,048 words) - 17:55, 19 March 2024
Divisible group (redirect from Injective group)
generated by injective modules is injective. The converse is a result of (Matlis 1958): if every module has a unique maximal injective submodule, then...
10 KB (1,411 words) - 20:34, 27 March 2024
field of category theory, the concept of injective object is a generalization of the concept of injective module. This concept is important in cohomology...
8 KB (1,031 words) - 17:57, 2 September 2022
injective hull) is a maximal essential extension, or a minimal embedding in an injective module. 3. An injective cogenerator is an injective module such...
20 KB (2,611 words) - 13:40, 22 July 2024
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...
20 KB (2,773 words) - 10:09, 18 February 2024
many of their desirable properties. Injective Injective modules are defined dually to projective modules. Flat A module is called flat if taking the tensor...
21 KB (2,946 words) - 19:00, 6 October 2024
Resolution (algebra) (redirect from Injective resolution)
projective modules or flat modules. Similarly every module has injective resolutions, which are right resolutions consisting of injective modules. Given a...
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Differential module Five lemma Short five lemma Snake lemma Nine lemma Extension (algebra) Central extension Splitting lemma Projective module Injective module Projective...
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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...
15 KB (2,210 words) - 17:59, 2 October 2024
Essential extension (category Module theory)
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...
6 KB (975 words) - 03:52, 29 July 2024
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...
23 KB (3,082 words) - 06:34, 3 October 2024
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...
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_{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...
30 KB (4,590 words) - 03:05, 9 August 2024
{\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...
4 KB (690 words) - 16:33, 17 November 2020
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...
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Localization (commutative algebra) (redirect from Localization of a module)
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...
29 KB (5,283 words) - 07:02, 25 July 2024
List of abstract algebra topics (section Module theory)
Projective module Projective cover Swan's theorem Quillen–Suslin theorem Injective module Injective hull Flat module Flat cover Coherent module Finitely-generated...
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Associated prime (redirect from Coprimary module)
coprimary modules. For a one-sided Noetherian ring, there is a surjection from the set of isomorphism classes of indecomposable injective modules onto the...
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German mathematician, known for his work in algebra. He introduced injective modules in 1940. He is the eponym of Baer rings and Baer groups. Baer studied...
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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...
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envelope or injective hull of a module is a smallest injective module containing it. 3. An injective resolution is a resolution by injective modules. 4. The...
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ideal Hilbert's Nullstellensatz Flat module Flat map Flat map (ring theory) Projective module Injective module Cohen-Macaulay ring Gorenstein ring Complete...
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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...
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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;...
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Endomorphism ring (category Module theory)
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...
9 KB (1,208 words) - 00:28, 6 March 2024
right) modules over the same ring, and let f : M → N be a module homomorphism. If M is simple, then f is either the zero homomorphism or injective because...
9 KB (1,345 words) - 07:14, 10 May 2024
{\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...
10 KB (1,249 words) - 15:50, 18 September 2024
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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