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
StatementIf 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 Classification
Statement Form Statement Form Connections
Has Statement Form
In Branch
Source Krull's principal ideal theorem (Wikipedia)
Principal Result Of
Source Krull's principal ideal theorem (Wikipedia)
Proved By
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.
Principal Result Of: Wolfgang Krull, Lead sentence of the article on the result
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 a commutative Noetherian ring.
- 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
In Group: Ring Theory and Commutative Algebra, Lead section
In commutative algebra, Krull's principal ideal theorem, named after
View the Source2. Wikidata: Krull's Principal Ideal Theorem
Reader Challenges (0)
No disputes yet. Spotted an error or a better source? Open the first one.
Sign in to dispute this or suggest a correction.