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Home > Krull's principal ideal theorem

 
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 Noetherian ring. The theorem is sometimes referred to by its German name, Krulls Hauptidealsatz.

Formally, if R is a Noetherian ring and I is a principal ideal of R, then I has height one.

This theorem can be generalized to ideals which are not principal, and the result is often called Krull's height theorem. It says, if R is a Noetherian ring and I is an ideal generated by n elements of R, then I has height at most n.


Commutative algebra Theorems

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Topics: Krull's Principal Ideal Theorem, Krull Dimension...

Krull dimensionCommutative algebra In commutative algebra, the Krull dimension of a ring R named after Wolfgang Krull (1899 1971), is defined to be the number of strict inclusions in a maximal chain of prime ideals. We take the supremum of chain lengths if no maximal chKrull's principal ideal theoremIn 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 Noetherian ring. The theorem is sometimes referred to by its German name, Krulls Hauptidealsatz''. FormalKrullKrull ( 1983) is a heroic fantasy film directed by Peter Yates and produced by Ron Silverman. It stars Ken Marshall as Prince Colwyn and Lysette Anthony as Princess Lyssa. One of the film's most distinguished features is the epic score by James Horner.
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