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