Commutative algebra Chain conditions Concept
Noetherian ring
Summary
A Noetherian ring is a ring in which every ideal is finitely generated, or equivalently, every increasing chain of ideals is eventually constant. The definition of a Noetherian ring gives the conditions.
Where this leads
49 pages build on this one.
- Betti numbers of a module
- Regular local rings are UFDs
- Krull intersection theorem
- Direct limit of rings
- Hochster's homological conjectures
- Dedekind domain
- Hilbert–Samuel function
- Matlis duality
- Artin–Rees lemma
- Excellent ring
- Goldie ring
- Local cohomology
- Artin–Tate lemma
- Noetherian scheme
- Complete local ring
- Serre's theorem on regular local rings
- Hilbert–Burch theorem
- Multiplicity of a local ring
- Embedding dimension
- Complete intersection ring
- Localization of a Noetherian ring is Noetherian
- Cohen–Macaulay ring
- Direct summand conjecture
- The local ring of holomorphic germs is Noetherian and factorial
- Characterizations of discrete valuation rings
- Gorenstein ring
- Kunz's theorem
- Posner's theorem
- Regular local ring
- Dimension theorem for local rings
- Goldie's theorem
- Depth
- Completion is flat
- Universally catenary ring
- Levitzki's theorem
- Lenagan's theorem
- Complex analytic space
- Unique factorization of ideals in a Dedekind domain
- Auslander–Buchsbaum formula
- Dimension of a polynomial ring
- Regular local rings are domains
- Serre's criterion for normality
- Hopkins–Levitzki theorem
- Nagata ring
- System of parameters
- Tight closure
- Nagata's example of a Noetherian ring of infinite dimension
- A Dedekind UFD is a PID
- Finitely presented module
Further reading
- Web https://en.wikipedia.org/wiki/Noetherian_ring, lead section
About this page
| Tag | 0670T59HGE |
|---|---|
| URI | m2://0670T59HGE |
| Kind | concept |
| Writing | W1 stated |
| MSC | 13E05 |
| Specializes | Ring |
| Same as |
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What this page needs for W2
- edge instance_of in: at least two examples (0 of 2)
- 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
Definitions
Used by
- Betti numbers of a module
- Krull intersection theorem
- Direct limit of rings
- Hochster's homological conjectures
- Dedekind domain
- Hilbert–Samuel function
- Matlis duality
- Artin–Rees lemma
- Excellent ring
- Goldie ring
- Local cohomology
- Artin–Tate lemma
- Noetherian scheme
- Complete local ring
- Serre's theorem on regular local rings
- Hilbert–Burch theorem
- Multiplicity of a local ring
- Embedding dimension
- Complete intersection ring
- Localization of a Noetherian ring is Noetherian
- Cohen–Macaulay ring
- Direct summand conjecture
- The local ring of holomorphic germs is Noetherian and factorial
- Characterizations of discrete valuation rings
- Gorenstein ring
- Kunz's theorem
- Regular local ring
- Dimension theorem for local rings
- Goldie's theorem
- Depth
- Completion is flat
- Universally catenary ring
- Levitzki's theorem
- Complex analytic space
- Unique factorization of ideals in a Dedekind domain
- Auslander–Buchsbaum formula
- Dimension of a polynomial ring
- Serre's criterion for normality
- Hopkins–Levitzki theorem
- Nagata ring
- System of parameters
- Tight closure
- Nagata's example of a Noetherian ring of infinite dimension
- Finitely presented module