Krull's theorem

Krull's theorem states that every nonzero commutative ring with identity has a maximal ideal. It follows by applying Zorn's lemma to the proper ideals ordered by inclusion.

Zorn's lemma: A principle stating that a partially ordered set in which every chain has an upper bound contains a maximal element. Applied to proper ideals, it supplies the maximal element the theorem asserts exists.
Commutative ring: A ring in which multiplication is commutative. Commutativity is part of the theorem's stated setting.
Wolfgang Krull: A German mathematician whose work helped establish modern abstract algebra and ideal theory. Krull's name became attached to this existence theorem in commutative algebra.
Prime ideal: A proper ideal whose quotient ring is an integral domain. Every maximal ideal in a commutative ring with identity is prime.
Maximal element: An element of a partially ordered set with no strictly greater element. The theorem yields an ideal maximal among proper ideals, not a greatest ideal containing all others.
Partially ordered set: A set equipped with a reflexive, antisymmetric, transitive relation. Ideal inclusion gives the ordering to which the maximal-element argument applies.
Ring with identity: A ring containing a multiplicative identity element. The theorem assumes an identity, which is used in standard ideal and quotient-ring formulations.
Ideal theory: The study of ideals and their roles in the structure of rings. Krull's theorem is a foundational existence result in the study of ring ideals.
Spectrum of a ring: The set of prime ideals of a commutative ring, commonly equipped with the Zariski topology. The theorem ensures this spectrum is nonempty for every nonzero commutative ring with identity.
Greatest element: An element of a partially ordered set greater than or equal to every element. A maximal ideal need not contain every proper ideal; confusing maximal with greatest overstates the conclusion.

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