Commutative algebra Basic notions Concept
Prime ideal
Summary
A prime ideal of a commutative ring is a proper ideal with the following property. If it contains a product, then it contains one of the factors. The prime ideals of are the ideals for prime and the zero ideal. The definition of a prime ideal gives the conditions.
Where this leads
38 pages build on this one.
- Jacobson ring
- Dedekind domain
- Discrete valuation ring
- Splitting of prime ideals
- Support of a module
- Cohen–Macaulay ring
- Ramification
- Irreducible analytic germ
- Zariski topology
- Local property
- Gorenstein ring
- Kunz's theorem
- Semistable reduction theorem
- Going-up theorem
- Jacobian criterion
- Normal ring
- Catenary ring
- Place (number field)
- Determinantal ideal
- Decomposition group
- Localization at a prime ideal
- Sober space
- Basic open set
- Incomparability theorem
- Spectrum of a ring
- Unramified morphism of rings
- Unique factorization of ideals in a Dedekind domain
- Dimension of a polynomial ring
- Proj construction
- Serre's criterion for normality
- Prime ideals of a localization
- Nagata ring
- Toric ideal
- Affine variety
- Going-down theorem
- Nagata's example of a Noetherian ring of infinite dimension
- A Dedekind UFD is a PID
- Lying over theorem
Further reading
- Web https://en.wikipedia.org/wiki/Prime_ideal, lead section
About this page
| Tag | 00MHHZ9SNE |
|---|---|
| URI | m2://00MHHZ9SNE |
| Kind | concept |
| Writing | W1 stated |
| MSC | 13A15 |
| Specializes | Ideal |
| 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)
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What links here
Definitions
Special cases
Used by
- Jacobson ring
- Dedekind domain
- Discrete valuation ring
- Splitting of prime ideals
- Support of a module
- Cohen–Macaulay ring
- Ramification
- Irreducible analytic germ
- Zariski topology
- Local property
- Gorenstein ring
- Kunz's theorem
- Going-up theorem
- Jacobian criterion
- Normal ring
- Catenary ring
- Place (number field)
- Determinantal ideal
- Decomposition group
- Localization at a prime ideal
- Sober space
- Basic open set
- Incomparability theorem
- Spectrum of a ring
- Unramified morphism of rings
- Unique factorization of ideals in a Dedekind domain
- Dimension of a polynomial ring
- Proj construction
- Serre's criterion for normality
- Prime ideals of a localization
- Nagata ring
- Toric ideal
- Affine variety
- Going-down theorem
- Nagata's example of a Noetherian ring of infinite dimension
- A Dedekind UFD is a PID
- Lying over theorem