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Every nontrivial unital commutative ring has a maximal ideal (via Zorn's lemma).
Existence of maximal ideals (Zorn): Let be a unital ring with , assumed commutative. Then has a maximal ideal.
This result is typically proved using Zorn's lemma (and hence the axiom of choice) applied to the partially ordered set of proper ideals of ordered by inclusion.
Let be a commutative ring with . A maximal ideal is a proper ideal such that there is no ideal with .
Zorn's lemma: Let be a partially ordered set. If every chain has an upper bound in , then has a maximal element.
Here a chain means a subset on which the restriction of is a total order, and an element is maximal if there is no with (i.e., and ).
Axiom of Choice: For every indexed family of nonempty sets, there exists a function such that for all .