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. The theorem is sometimes referred to by its German name, Krulls Hauptidealsatz (Satz meaning "proposition" or "theorem"). Precisely, if R is a Noetherian ring and I is a principal, proper ideal of R, then each minimal prime ideal over I has height at most one.
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| - Krullscher Hauptidealsatz (de)
- Théorème des idéaux principaux de Krull (fr)
- Teorema dell'ideale principale (it)
- Krull's principal ideal theorem (en)
- 크룰 높이 정리 (ko)
- クルルの単項イデアル定理 (ja)
- Теорема Круля про головний ідеал (uk)
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| - Der Krullsche Hauptidealsatz ist ein zentraler Satz der Dimensionstheorie von noetherschen Ringen in der kommutativen Algebra, der nach Wolfgang Krull benannt ist und von ihm 1928 veröffentlicht wurde. (de)
- En algèbre commutative, le théorème des idéaux principaux de Krull (Krulls Hauptidealsatz) est un résultat fondamental en théorie de la dimension. Intuitivement, il dit grosso modo qu'une hypersurface est de codimension 1. (fr)
- 可換環論（次元論）において、クルルの単項イデアル定理（英: Krull's principal ideal theorem, Krulls Hauptidealsatz）は、ネーター環の素イデアルの高さについての基本的な定理である。 (ja)
- 가환대수학에서 크룰 높이 정리(영어: Krull’s height theorem)는 뇌터 환에서 n개의 원소로 생성된 아이디얼의 높이가 n 이하라는 정리이다. (ko)
- In matematica, il teorema dell'ideale principale (a volte citato, in tedesco, come Hauptidealsatz) è un teorema di algebra commutativa che stabilisce un'importante proprietà degli anelli commutativi noetheriani. È stato dimostrato da Wolfgang Krull nel 1928. (it)
- Теорема Круля про головний ідеал — важливе твердження у комутативній алгебрі, яке разом зі своїми наслідками є основою для означення розмірності в алгебрі і алгебричній геометрії. Теорема названа на честь австрійського математика Вольфганга Круля. (uk)
- 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. The theorem is sometimes referred to by its German name, Krulls Hauptidealsatz (Satz meaning "proposition" or "theorem"). Precisely, if R is a Noetherian ring and I is a principal, proper ideal of R, then each minimal prime ideal over I has height at most one. (en)
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| - Der Krullsche Hauptidealsatz ist ein zentraler Satz der Dimensionstheorie von noetherschen Ringen in der kommutativen Algebra, der nach Wolfgang Krull benannt ist und von ihm 1928 veröffentlicht wurde. (de)
- 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. The theorem is sometimes referred to by its German name, Krulls Hauptidealsatz (Satz meaning "proposition" or "theorem"). Precisely, if R is a Noetherian ring and I is a principal, proper ideal of R, then each minimal prime ideal over I has height at most one. This theorem can be generalized to ideals 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. The principal ideal theorem and the generalization, the height theorem, both follow from the fundamental theorem of dimension theory in commutative algebra (see also below for the direct proofs). Bourbaki's Commutative Algebra gives a direct proof. Kaplansky's Commutative Rings includes a proof due to David Rees. (en)
- En algèbre commutative, le théorème des idéaux principaux de Krull (Krulls Hauptidealsatz) est un résultat fondamental en théorie de la dimension. Intuitivement, il dit grosso modo qu'une hypersurface est de codimension 1. (fr)
- 可換環論（次元論）において、クルルの単項イデアル定理（英: Krull's principal ideal theorem, Krulls Hauptidealsatz）は、ネーター環の素イデアルの高さについての基本的な定理である。 (ja)
- 가환대수학에서 크룰 높이 정리(영어: Krull’s height theorem)는 뇌터 환에서 n개의 원소로 생성된 아이디얼의 높이가 n 이하라는 정리이다. (ko)
- In matematica, il teorema dell'ideale principale (a volte citato, in tedesco, come Hauptidealsatz) è un teorema di algebra commutativa che stabilisce un'importante proprietà degli anelli commutativi noetheriani. È stato dimostrato da Wolfgang Krull nel 1928. (it)
- Теорема Круля про головний ідеал — важливе твердження у комутативній алгебрі, яке разом зі своїми наслідками є основою для означення розмірності в алгебрі і алгебричній геометрії. Теорема названа на честь австрійського математика Вольфганга Круля. (uk)
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