# Krull's theorem

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, 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.
- A 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*.

- Let

- This result implies the original theorem, 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*.

## 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.

## Notes

- In this article, rings have a 1.

## References

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

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