Krull's Theorem - Krull's Theorem

Krull's Theorem

Let R be a commutative 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 commutative 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.