Krull's theorem

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,[2] 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.
  • 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 article: Krull's principal ideal theorem

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
  1. In this article, rings have a 1.
  2. Hodges, W. (1979). "Krull implies Zorn". Journal of the London Mathematical Society. s2-19 (2): 285–287. doi:10.1112/jlms/s2-19.2.285.

References
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