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Canonical static concept record

Prime ideal

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Summary

A proper ideal $P$ of a commutative ring such that $ab\in P$ implies $a\in P$ or $b\in P$ ; equivalently, the quotient $R/P$ is an integral domain.

Record metadata

Canonical name: Prime ideal
Concept ID: prime_ideal
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Number theory
Domains: Number theory, Algebra, Algebraic geometry
Alternate terminology: No alternate terminology is currently authored for this record.
Concept type: Prime ring ideal

Concept sources

Milne — Algebraic Number Theory — Algebraic Number Theory · course text · source ID milne-algebraic-number-theory
Stacks Project — Commutative Algebra (PDF) — Commutative Algebra (PDF) · reference text · source ID stacks-algebra-text

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Ideal Prime ideal

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: number_ideal_to_prime_ideal
Relation type: Classification classification
Direction: source → target
Endpoint roles: source: Has subtype; target: Classified as
Authored annotation: satisfies primality

Authored explanation

Prime ideals are the proper ideals for which a product can enter the ideal only through at least one factor, a condition equivalent in commutative rings to domain-valued quotient.

How to interpret this relation type

The target is a member or subtype of the broader source class.

Relation sources

Milne — Algebraic Number Theory — Algebraic Number Theory · course text · source ID milne-algebraic-number-theory
Stacks Project — Commutative Algebra (PDF) — Commutative Algebra (PDF) · reference text · source ID stacks-algebra-text

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.

Prime ideal Prime spectrum $\operatorname{Spec}(R)$

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: number_prime_ideal_to_prime_spectrum
Relation type: Canonical construction canonical-construction
Direction: source → target
Endpoint roles: source: Canonically constructs; target: Canonically constructed from
Authored annotation: supply points of $\operatorname{Spec}(R)$

Authored explanation

The prime spectrum of a commutative ring is formed from its prime ideals, equipped with the Zariski topology and its canonical structure sheaf.

How to interpret this relation type

Apply a standard functorial or canonical construction whose output is not merely a reduct of the input and is not generally an equivalent presentation of the same object.

Relation sources

Stacks Project — Commutative Algebra (PDF) — Commutative Algebra (PDF) · reference text · source ID stacks-algebra-text
Stacks Project — Schemes — Schemes · reference · source ID stacks-scheme

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Prime ideal

Summary

Record metadata

Concept sources

Incoming relations (arrows to this concept)

Ideal Prime ideal

Authored explanation

How to interpret this relation type

Relation sources

Outgoing relations (arrows from this concept)

Prime ideal Prime spectrum $\operatorname{Spec}(R)$

Authored explanation

How to interpret this relation type

Relation sources