Krull's theorem

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In mathematics, more specifically in ring theory, Krull's theorem, named after Wolfgang Krull, proves the existence of maximal ideals in any unital commutative ring. The theorem was first stated in 1929 and is equivalent to the axiom of choice.

[edit] Krull's theorem

Let R be a unital commutative ring, which is not the trivial ring. Then R contains a maximal ideal.

The statement can be proved using Zorn's lemma, which in turn requires (or rather is equivalent to) the axiom of choice.

A slightly stronger result, which can be proved in a similar fashion, is as follows: Let R be a unital commutative 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).

[edit] 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.