Commutative algebra Local rings Concept
Local ring
Summary
A local ring is a commutative ring with exactly one maximal ideal. Equivalently, its non-units form an ideal. Local rings arise as localizations at prime ideals and describe the behaviour of a ring, or of a geometric space, near one point. Many theorems of commutative algebra are first proved for local rings, where Nakayama's lemma is a strong tool.
Where this leads
31 pages build on this one.
- Betti numbers of a module
- Nakayama's lemma
- Regular local rings are UFDs
- Krull intersection theorem
- Hochster's homological conjectures
- Hilbert–Samuel function
- Complete local ring
- Residue field
- Serre's theorem on regular local rings
- Finitely generated projective modules over local rings are free
- Multiplicity of a local ring
- Embedding dimension
- Complete intersection ring
- Cohen–Macaulay ring
- Strict Henselization
- Characterizations of discrete valuation rings
- Canonical module
- Gorenstein ring
- Structure theorem for Artinian rings
- Henselization
- Regular local ring
- Dimension theorem for local rings
- Localization at a prime ideal
- Depth
- Local homomorphism
- Auslander–Buchsbaum formula
- Cotangent space of a local ring
- Regular local rings are domains
- Grothendieck's local duality
- Henselian ring
- System of parameters
Further reading
About this page
| Tag | MZ02BKPHJA |
|---|---|
| URI | m2://MZ02BKPHJA |
| Kind | concept |
| Writing | W0 stub |
| MSC | 13H99, 13B30 |
| Specializes | Commutative ring |
| Same as | |
| Changed | 2 November 2026 |
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What links here
Used by
- Betti numbers of a module
- Nakayama's lemma
- Krull intersection theorem
- Hochster's homological conjectures
- Hilbert–Samuel function
- Complete local ring
- Residue field
- Serre's theorem on regular local rings
- Finitely generated projective modules over local rings are free
- Multiplicity of a local ring
- Embedding dimension
- Complete intersection ring
- Cohen–Macaulay ring
- Strict Henselization
- Characterizations of discrete valuation rings
- Canonical module
- Gorenstein ring
- Structure theorem for Artinian rings
- Henselization
- Regular local ring
- Dimension theorem for local rings
- Localization at a prime ideal
- Depth
- Local homomorphism
- Auslander–Buchsbaum formula
- Cotangent space of a local ring
- Grothendieck's local duality
- Henselian ring
- System of parameters