Krull's theorem
In mathematics, more specifically in ring theory, Krull's theorem, named after Wolfgang Krull, proves the existence of maximal ideals in any unital ring. The theorem was first stated in 1929 and is equivalent to the axiom of choice.
Krull's theorem
Let R be a unital ring, which is not the trivial ring. Then R contains a maximal ideal.
The statement can be proved using Zorn's lemma, which is equivalent to the axiom of choice.
A slightly stronger (but equivalent) result, which can be proved in a similar fashion, is as follows: Let R be a unital ring which is not the trivial ring, and let I be a proper ideal of R. Then there is a maximal ideal of R containing I. Note that this result does indeed imply the previous theorem, by taking I to be the zero ideal (0).
Krull's Hauptidealsatz
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.
References
- W. Krull, Die Idealtheorie in Ringen ohne Endlicheitsbedingungen, Mathematische Annalen 10 (1929), 729–744.