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Krull's Principal Ideal Theorem


The most general form of this theorem states that in a commutative unit ring R, the height of every proper ideal I generated by n elements is at most n. Equality is attained if these n elements form a regular sequence.

Setting n=1 yields part of the original statement on principal ideals, also known under the German name Hauptidealsatz, that for every nonzero, noninvertible element a of R, the ideal I=<a> of R has height at most 1, and, moreover, heightI=1 iff a is a non-zero divisor.

It immediately follows as a corollary that every proper ideal of a Noetherian ring has finite height and that a principal ideal domain has Krull dimension equal to 1.


See also

Krull Dimension, Principal Ideal

This entry contributed by Margherita Barile

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References

Matsumura, H. Commutative Ring Theory. Cambridge, England: Cambridge University Press, p. 100, 1986.

Referenced on Wolfram|Alpha

Krull's Principal Ideal Theorem

Cite this as:

Barile, Margherita. "Krull's Principal Ideal Theorem." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/KrullsPrincipalIdealTheorem.html

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