Krull's theorem

In mathematics, and more specifically in ring theory, Krull's theorem, named after Wolfgang Krull, asserts that a nonzero ring 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

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 result, which can be proved in a similar fashion, is as follows: