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The | In [[mathematics]], and more specifically in [[ring theory]], '''Krull's theorem''', named after [[Wolfgang Krull]], asserts that a [[zero ring|nonzero]] [[ring_(mathematics)|ring]]<ref>In this article, rings have a 1.</ref> has at least one [[maximal ideal]]. The theorem was proved in 1929 by Krull, who used [[transfinite induction]]. | ||
The theorem admits a [[Zorn's lemma#An example application|simple proof using Zorn's lemma]], and in fact is equivalent to [[Zorn's lemma]], | |||
which in turn is equivalent to the [[axiom of choice]]. | |||
==Variants== | |||
* For [[noncommutative ring]]s, the analogues for maximal left ideals and maximal right ideals also hold. | |||
* For [[pseudo-ring]]s, the theorem holds for [[regular ideal]]s.{{disambiguate|date=December 2013}} | |||
* A slightly stronger (but equivalent) result, which can be proved in a similar fashion, is as follows: | |||
:::Let ''R'' be a ring, and let ''I'' be a [[proper ideal]] of ''R''. Then there is a maximal ideal of ''R'' containing ''I''. | |||
:This result implies the original theorem, by taking ''I'' to be the [[zero ideal]] (0). Conversely, applying the original theorem to ''R''/''I'' leads to this result. | |||
:To prove the stronger result directly, consider the set ''S'' of all proper ideals of ''R'' containing ''I''. The set ''S'' is nonempty since ''I'' ∈ ''S''. Furthermore, for any chain ''T'' of ''S'', the union of the ideals in ''T'' is an ideal ''J'', and a union of ideals not containing 1 does not contain 1, so ''J'' ∈ ''S''. By Zorn's lemma, ''S'' has a maximal element ''M''. This ''M'' is a maximal ideal containing ''I''. | |||
== Krull's Hauptidealsatz == | |||
{{main|Krull's principal ideal theorem}} | |||
Another theorem commonly referred to as Krull's theorem: | |||
:::Let <math>R</math> be a Noetherian ring and <math>a</math> an element of <math>R</math> which is neither a [[zero divisor]] nor a [[Unit (ring theory)|unit]]. Then every minimal [[prime ideal]] <math>P</math> containing <math>a</math> has [[height (ring theory)|height]] 1. | |||
==Notes== | |||
{{reflist}} | |||
== References == | |||
* W. Krull, ''Idealtheorie in Ringen ohne Endlichkeitsbedingungen'', [[Mathematische Annalen]] '''10''' (1929), 729–744. | |||
[[Category:Ideals]] | |||
Latest revision as of 08:24, 15 March 2013
In mathematics, and more specifically in ring theory, Krull's theorem, named after Wolfgang Krull, asserts that a nonzero ring[1] has at least one maximal ideal. The theorem was proved in 1929 by Krull, who used transfinite induction. The theorem admits a simple proof using Zorn's lemma, and in fact is equivalent to Zorn's lemma, which in turn is equivalent to the axiom of choice.
Variants
- For noncommutative rings, the analogues for maximal left ideals and maximal right ideals also hold.
- For pseudo-rings, the theorem holds for regular ideals.Template:Disambiguate
- A slightly stronger (but equivalent) result, which can be proved in a similar fashion, is as follows:
- Let R be a ring, and let I be a proper ideal of R. Then there is a maximal ideal of R containing I.
- This result implies the original theorem, by taking I to be the zero ideal (0). Conversely, applying the original theorem to R/I leads to this result.
- To prove the stronger result directly, consider the set S of all proper ideals of R containing I. The set S is nonempty since I ∈ S. Furthermore, for any chain T of S, the union of the ideals in T is an ideal J, and a union of ideals not containing 1 does not contain 1, so J ∈ S. By Zorn's lemma, S has a maximal element M. This M is a maximal ideal containing I.
Krull's Hauptidealsatz
Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. Another theorem commonly referred to as Krull's theorem:
- Let be a Noetherian ring and an element of which is neither a zero divisor nor a unit. Then every minimal prime ideal containing has height 1.
Notes
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References
- W. Krull, Idealtheorie in Ringen ohne Endlichkeitsbedingungen, Mathematische Annalen 10 (1929), 729–744.
- ↑ In this article, rings have a 1.