Editing Krull's principal ideal theorem (section)

{{short description|Theorem in commutative algebra}}
In [[commutative algebra]], '''Krull's principal ideal theorem''', named after [[Wolfgang Krull]] (1899–1971),  gives a bound on the [[height (ring theory)|height]] of a [[principal ideal]] in a commutative [[Noetherian ring]].  The theorem is sometimes referred to by its German name, ''Krulls Hauptidealsatz'' (from ''{{Wikt-lang|en|Haupt-}}'' ("Principal") + ''{{Wikt-lang|en|ideal}}'' +  ''{{Wikt-lang|en|Satz}}'' ("theorem")).

Precisely, 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.

This theorem can be generalized to [[ideal (ring theory)|ideal]]s that are not principal, and the result is often called '''Krull's height theorem'''.  This says that if ''R'' is a Noetherian ring and ''I'' is a proper ideal generated by ''n'' elements of ''R'', then each minimal prime over ''I'' has height at most ''n''. The converse is also true: if a prime ideal has height ''n'', then it is a minimal prime ideal over an ideal generated by ''n'' elements.<ref>{{harvnb|Eisenbud|1995|loc=Corollary 10.5.}}</ref>

The principal ideal theorem and the generalization, the height theorem, both follow from the [[fundamental theorem of dimension theory (algebra)|fundamental theorem of dimension theory]] in commutative algebra (see also below for the direct proofs). Bourbaki's ''[[Éléments de mathématique|Commutative Algebra]]'' gives a direct proof. Kaplansky's ''Commutative Rings'' includes a proof due to [[David Rees (mathematician)|David Rees]].