• 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...
    6 KB (757 words) - 18:08, 23 May 2023
  • 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
  • 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
  • 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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  • Thumbnail for Bijection, injection and surjection
    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...
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  • {\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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  • 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...
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  • Projective module Projective cover Swan's theorem Quillen–Suslin theorem Injective module Injective hull Flat module Flat cover Coherent module Finitely-generated...
    12 KB (1,128 words) - 01:18, 14 November 2023
  • 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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  • Thumbnail for Reinhold Baer
    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...
    66 KB (9,767 words) - 00:23, 7 July 2024
  • ideal Hilbert's Nullstellensatz Flat module Flat map Flat map (ring theory) Projective module Injective module Cohen-Macaulay ring Gorenstein ring Complete...
    4 KB (301 words) - 17:28, 20 December 2023
  • 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...
    4 KB (513 words) - 07:01, 23 May 2021
  • 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...
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  • {\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...
    1 KB (172 words) - 15:16, 2 February 2024