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Krull's theorem

Part of ring theory in mathematics

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 (but equivalent) result, which can be proved in a similar fashion, is as follows:

: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 ## References - - ## References 1. In this article, rings have a 1. 2. Hodges, W.. (1979). "Krull implies Zorn". *[[Journal of the London Mathematical Society]]*. ::callout[type=info title="Wikipedia Source"] This article was imported from [Wikipedia](https://en.wikipedia.org/wiki/Krull's_theorem) and is available under the [Creative Commons Attribution-ShareAlike 4.0 License](https://creativecommons.org/licenses/by-sa/4.0/). Content has been adapted to SurfDoc format. Original contributors can be found on the [article history page](https://en.wikipedia.org/wiki/Krull's_theorem?action=history). ::

ideals-(ring-theory) theorems-in-algebra

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