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A commutative ring with 1 and 1≠0 contains at least one maximal ideal.
Every nontrivial commutative ring has a maximal ideal: If is a commutative ring with and , then there exists a maximal ideal .
This follows from the existence theorem for maximal ideals, whose proof applies Zorn's Lemma to the set of proper ideals of a commutative ring with to produce a maximal ideal.
Existence of maximal ideals (Zorn): Let be a unital ring with , assumed commutative. Then has a maximal ideal.
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 ).
An ideal in a ring is an additive subgroup such that it is stable under multiplication by elements of on one side:
Let be a commutative ring with . A maximal ideal is a proper ideal such that there is no ideal with .