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
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Source Krull's Theorem (Wikipedia)
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.
Proved By: Wolfgang Krull, Lead
The theorem was proved in 1929 by Krull, who used transfinite induction.
In Group: Ring Theory and Commutative Algebra, Lead section
In mathematics, and more specifically in ring theory, Krull's theorem, named after Wolfgang Krull
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