Commutative algebra Local rings Concept
Regular local ring
Summary
A regular local ring is a Noetherian local ring whose maximal ideal can be generated by as many elements as its Krull dimension. Equivalently, its embedding dimension equals its dimension. Regular local rings are the algebraic form of smooth points of varieties. They are integral domains and even unique factorization domains, and by Serre's theorem they are the Noetherian local rings of finite global dimension.
Where this leads
12 pages build on this one.
- Regular local rings are UFDs
- Hochster's homological conjectures
- Excellent ring
- Serre's theorem on regular local rings
- Complete intersection ring
- Direct summand conjecture
- Characterizations of discrete valuation rings
- Kunz's theorem
- Jacobian criterion
- Cohen structure theorem
- Regular local rings are domains
- Serre's criterion for normality
Further reading
About this page
| Tag | 63XXTF6B5M |
|---|---|
| URI | m2://63XXTF6B5M |
| Kind | concept |
| Writing | W0 stub |
| MSC | 13H05 |
| Specializes | Local ring |
| Same as | |
| Changed | 2 November 2026 |
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Results about this
- Regular local rings are UFDs
- Hochster's homological conjectures
- Serre's theorem on regular local rings
- Direct summand conjecture
- Kunz's theorem
- Jacobian criterion
- Regular local rings are domains
Used by
- Regular local rings are UFDs
- Hochster's homological conjectures
- Excellent ring
- Serre's theorem on regular local rings
- Complete intersection ring
- Direct summand conjecture
- Characterizations of discrete valuation rings
- Kunz's theorem
- Jacobian criterion
- Cohen structure theorem
- Regular local rings are domains
- Serre's criterion for normality