Krull's Theorem states that every nonzero ring with identity has at least one maximal ideal, and more generally that every proper ideal of such a ring is contained in some maximal ideal. Named for Wolfgang Krull, its standard proof rests on Zorn's Lemma, and it is a basic existence result underlying the construction of quotient fields and the theory of local rings.
Facts
StatementEvery ring with a multiplicative identity, other than the zero ring, has at least one maximal ideal, and every proper ideal in such a ring is contained in some maximal ideal. 1 Proof YearWolfgang Krull proved the theorem in 1929 using transfinite induction. It is equivalent to Zorn's lemma and to the axiom of choice. Classification
Statement Form Connections
Has Statement Form
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
In Branch
Sources
1. Krull's Theorem (Wikipedia)
Wikimedia FoundationLead section, second sentence
The theorem was proved in 1929 by Krull, who used transfinite induction.
Lead section, statement-form reference
In mathematics, and more specifically in ring theory, Krull's theorem, named after Wolfgang Krull, asserts that a nonzero ring has at least one maximal ideal.
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