# 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 commutative unital ring. The theorem was first stated in 1929 and is equivalent to the axiom of choice.

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). To prove the statement, consider the set S of all proper ideals of R containing I. S is certainly nonempty as I is an element of S. Furthermore, for any chain T of S, it is easy to see that union of ideals in T is an ideal J. In fact, J is proper (otherwise 1J implying that 1N for some NT contradicting that TS). Therefore by Zorn's lemma, S has a maximal element which must be a maximal ideal containing I.

For non-unital rings, the theorem holds for regular ideals.

## Krull's Hauptidealsatz

Another theorem commonly referred to as Krull's theorem: Let $R$ be a Noetherian ring and $a$ an element of $R$ which is neither a zero divisor nor a unit. Then every minimal prime ideal $P$ containing $a$ has height 1.

## References

• W. Krull, Idealtheorie in Ringen ohne Endlichkeitsbedingungen, Mathematische Annalen 10 (1929), 729–744.