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Theorem

Krull's Principal Ideal Theorem

Algebra

Krull's Principal Ideal Theorem states that in a Noetherian commutative ring, every prime ideal that is minimal over an ideal generated by a single element has height at most one, meaning no chain of prime ideals properly contained in it can have length greater than one. Named for Wolfgang Krull, it is a foundational result of commutative algebra controlling how the codimension of a subvariety can grow when it is cut out by a single equation, and it generalizes to a bound of at most n for an ideal generated by n elements.

Facts
Statement
If R is a Noetherian ring and I is a principal, proper ideal of R, then each minimal prime ideal containing I has height at most one. 1
Proof Year
1928 2
Classification
Statement Form
Uniqueness Theorem 1
Statement Form
Inequality 1
Connections

Has Statement Form

Inequality, Concepts

Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.

Uniqueness Theorem, Concepts

Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.

In Branch

Commutative Algebra, Branches of Mathematics
Source Krull's principal ideal theorem (Wikipedia)

Proved By

Wolfgang Krull, Mathematicians
Source Krull's principal ideal theorem (Wikipedia)
Sources
1. Krull's principal ideal theorem (Wikipedia)
  • Statement of the Theorem
    If R is a Noetherian ring and I is a principal, proper ideal of R, then each minimal prime ideal containing I has height at most one.
  • In Branch: Commutative Algebra, Lead sentence
  • Proved By: Wolfgang Krull, Lead paragraph
    In commutative algebra, Krull's principal ideal theorem, named after Wolfgang Krull (1899-1971), gives a bound on the height of a principal ideal in
View the Source
2. Wikidata: Krull's Principal Ideal Theorem
View the Source
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