Krull's theorem

Not to be confused with Krull's principal ideal theorem, a theorem on the height of ideals in a Noetherian Ring.

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.
• An apparently 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.

The statement of the original theorem can be obtained 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.

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

• Krull, W. (1929). "Idealtheorie in Ringen ohne Endlichkeitsbedingungen". Mathematische Annalen . 101 (1): 729–744. doi:10.1007/BF01454872. S2CID   119883473.
• 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.