Commutative algebra Basic notions Concept
Commutative ring
Summary
A commutative ring is a ring in which the multiplication is commutative. Commutative algebra is the study of these rings. The definition of a commutative ring gives the precise condition.
Where this leads
92 pages build on this one.
- Nakayama's lemma
- Elliptic cohomology
- Jacobson ring
- Koszul complex
- Temperley–Lieb algebra
- Étale morphism of rings
- Ringed space
- Vandermonde matrix
- Regular sequence
- Rees algebra
- Massey product
- Global dimension
- Complete local ring
- Sheaf
- Sylvester's determinant identity
- Flat module
- Ring of convergent power series
- Faithfully flat module
- Total ring of fractions
- Kaplansky's theorem on PI rings
- Cap product
- Formal group law in topology
- Finitely generated projective modules over local rings are free
- Linear topology
- Determinant trick
- Tensor-hom adjunction
- Hecke algebra
- Möbius inversion on posets
- Intersection multiplicity
- Kaplansky's theorem on projective modules
- Pfaffian
- Witt vectors
- Graded ring
- Exact sequence
- Formally smooth map
- Tensor product of algebras
- Projective module
- Affine scheme
- Bass's theorem P
- C-infinity ring
- Cohomology ring
- Zariski topology
- De Rham cohomology of varieties
- Exact couple
- Local property
- Posner's theorem
- Filtration of a module
- Cellular algebra
- Cluster algebra
- Normal ring
- Free algebra
- Serre's twisting sheaf
- Lindström–Gessel–Viennot lemma
- Catenary ring
- Multivariate formal power series
- Localization
- Determinantal ideal
- Crystalline cohomology
- I-adic topology
- Cohn localization
- Incidence algebra
- Completion is flat
- Completion of a ring
- Differential graded algebra
- Multiplicative set
- Semi-local ring
- Polynomial identity ring
- Tor functor
- Picard group of a ring
- Spectrum of a ring
- Permanent
- Localization of a space
- Unramified morphism of rings
- Injective module
- Kähler differentials
- Ring of invariants
- Cayley–Hamilton theorem for modules
- Tensor product of modules
- Dimension of a polynomial ring
- Ring of differential operators
- Bockstein homomorphism
- Torsion submodule
- Cup product
- Azumaya algebra
- Cobordism ring
- Local ring
- Synthetic differential geometry
- Integral element
- Associated graded ring
- Amitsur–Levitzki theorem
- Finitely presented module
- Smooth morphism of rings
Further reading
- Web https://en.wikipedia.org/wiki/Commutative_ring, lead section
About this page
| Tag | 0310REJJY8 |
|---|---|
| URI | m2://0310REJJY8 |
| Kind | concept |
| Writing | W1 stated |
| MSC | 13A99 |
| Specializes | Ring |
| Same as |
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What this page needs for W2
- edge not_instance_of in: at least one non-example (0 of 1)
The writing levels say how complete the text of a page is. They do not measure its truth.
What links here
Instances
Definitions
Special cases
- Integral domain
- Artinian ring
- Jacobson ring
- Catenary ring
- Commutative algebra over a ring
- Galois ring
- Local ring
- Reduced ring
- Cluster algebra
- Coherent ring
Used by
- Nakayama's lemma
- Elliptic cohomology
- Jacobson ring
- Koszul complex
- Temperley–Lieb algebra
- Étale morphism of rings
- Ringed space
- Vandermonde matrix
- Regular sequence
- Rees algebra
- Massey product
- Global dimension
- Sheaf
- Sylvester's determinant identity
- Flat module
- Faithfully flat module
- Total ring of fractions
- Cap product
- Formal group law in topology
- Linear topology
- Determinant trick
- Tensor-hom adjunction
- Hecke algebra
- Möbius inversion on posets
- Pfaffian
- Witt vectors
- Graded ring
- Exact sequence
- Formally smooth map
- Tensor product of algebras
- Projective module
- Affine scheme
- C-infinity ring
- Cohomology ring
- Zariski topology
- Local property
- Filtration of a module
- Cellular algebra
- Cluster algebra
- Normal ring
- Free algebra
- Lindström–Gessel–Viennot lemma
- Catenary ring
- Multivariate formal power series
- Localization
- Determinantal ideal
- I-adic topology
- Cohn localization
- Incidence algebra
- Completion of a ring
- Multiplicative set
- Semi-local ring
- Polynomial identity ring
- Tor functor
- Picard group of a ring
- Spectrum of a ring
- Permanent
- Unramified morphism of rings
- Injective module
- Kähler differentials
- Ring of invariants
- Cayley–Hamilton theorem for modules
- Tensor product of modules
- Dimension of a polynomial ring
- Ring of differential operators
- Torsion submodule
- Cup product
- Local ring
- Synthetic differential geometry
- Integral element
- Associated graded ring
- Amitsur–Levitzki theorem
- Finitely presented module
- Smooth morphism of rings