Krull's theorem

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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[edit]

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 IS. 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 JS. By Zorn's lemma, S has a maximal element M. This M is a maximal ideal containing I.

Krull's Hauptidealsatz[edit]

Another theorem commonly referred to as Krull's theorem:

Let R be a Noetherian ring and a an element of R which is neither a zero divisor nor a unit. Then every minimal prime ideal P containing a has height 1.

Notes[edit]

  1. ^ In this article, rings have a 1.

References[edit]

  • W. Krull, Idealtheorie in Ringen ohne Endlichkeitsbedingungen, Mathematische Annalen 10 (1929), 729–744.