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.
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 1 ∈ J implying that 1 ∈ N for some N ∈ T contradicting that T ⊂ S). 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.
- W. Krull, Idealtheorie in Ringen ohne Endlichkeitsbedingungen, Mathematische Annalen 10 (1929), 729–744.
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