Associative rings and algebras Modules and ideals Concept
Module over a ring
Summary
A module over a ring is an abelian group with a multiplication by the elements of the ring. This multiplication satisfies the same axioms as for a vector space.
Where this leads
79 pages build on this one.
- Additive category
- Nakayama's lemma
- Grothendieck category
- Koszul complex
- Serial ring
- Gabriel–Popescu theorem
- Hopf module
- Projective cover
- Auslander–Reiten theory
- Local cohomology
- Sheaf of modules
- Self-injective ring
- Regular sequence
- Projective dimension
- Global dimension
- Weak dimension
- Modular representation theory
- Splitting lemma
- Flat module
- Faithfully flat module
- Freyd–Mitchell embedding theorem
- Fractional ideal
- Finitely generated projective modules over local rings are free
- Injective hull
- Finite morphism of rings
- Linear topology
- Tate cohomology
- Abelian category
- Tensor-hom adjunction
- Support of a module
- Intersection multiplicity
- Kaplansky's theorem on projective modules
- Exact sequence
- Morita equivalence
- Projective module
- Bass's theorem P
- Ore extension
- Socle of a module
- Smash product
- Local coefficient system
- Eilenberg swindle
- De Rham cohomology of varieties
- Exact couple
- Local property
- Almost split sequence
- Ext functor
- Filtration of a module
- Category of modules
- Morita equivalence of a ring with its matrix rings
- Four lemma
- Progenerator
- Cokernel
- Krull dimension of a noncommutative ring
- I-adic topology
- Baer's criterion
- Semisimple ring
- Uniserial module
- Preadditive category
- Chain complex
- Group homology
- Quasi-coherent sheaf
- Tor functor
- Double complex
- Lenagan's theorem
- Projective resolution
- Injective module
- Kähler differentials
- Localization of a module
- Tensor product of modules
- Structure theorem for finitely generated modules over a principal ideal domain
- Radical of a module
- Bockstein homomorphism
- Torsion submodule
- Five lemma
- Invariant basis number
- Azumaya algebra
- Yoneda Ext and extensions
- Kernel
- Finitely presented module
Further reading
- Web https://en.wikipedia.org/wiki/Module_(mathematics), Formal definition
About this page
| Tag | 0NZ8JA25WK |
|---|---|
| URI | m2://0NZ8JA25WK |
| Kind | concept |
| Writing | W1 stated |
| MSC | 16D10, 13C05 |
| Same as |
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What this page needs for W2
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- edge not_instance_of in: at least one non-example (0 of 1)
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What links here
Special cases
- Vector space
- Finitely generated module
- Submodule
- Free module
- Flat module
- Injective module
- Noetherian module
- Projective module
- Artinian module
- Uniserial module
Definitions
Used by
- Additive category
- Nakayama's lemma
- Grothendieck category
- Koszul complex
- Serial ring
- Hopf module
- Projective cover
- Auslander–Reiten theory
- Local cohomology
- Sheaf of modules
- Self-injective ring
- Regular sequence
- Projective dimension
- Global dimension
- Weak dimension
- Modular representation theory
- Flat module
- Faithfully flat module
- Fractional ideal
- Injective hull
- Finite morphism of rings
- Linear topology
- Tate cohomology
- Abelian category
- Tensor-hom adjunction
- Support of a module
- Exact sequence
- Morita equivalence
- Projective module
- Ore extension
- Socle of a module
- Smash product
- Local coefficient system
- Eilenberg swindle
- Local property
- Almost split sequence
- Ext functor
- Filtration of a module
- Category of modules
- Morita equivalence of a ring with its matrix rings
- Progenerator
- Cokernel
- Krull dimension of a noncommutative ring
- I-adic topology
- Baer's criterion
- Semisimple ring
- Uniserial module
- Preadditive category
- Chain complex
- Group homology
- Quasi-coherent sheaf
- Tor functor
- Double complex
- Lenagan's theorem
- Projective resolution
- Injective module
- Kähler differentials
- Localization of a module
- Tensor product of modules
- Structure theorem for finitely generated modules over a principal ideal domain
- Radical of a module
- Torsion submodule
- Invariant basis number
- Yoneda Ext and extensions
- Kernel
- Finitely presented module