Linked from

The 15 pages that link to Maximal ideal, each with the reason it gives.

• Prime idealCompared with: Every maximal ideal in a commutative ring with identity is prime, but not conversely.

• Commutative ringRelated: Quotienting a commutative ring by a maximal ideal produces a field.

• Ideal (ring theory)Broader topic: It is distinguished by maximality rather than the product test defining prime ideals.

• Zorn's lemmaBroader topic: Applying Zorn's lemma to proper ideals proves that every nonzero unital commutative ring has a maximal ideal.

• IdealBroader topic: Maximal ideals classify residue fields and local algebraic behavior.

• Hilbert's NullstellensatzRelated: The weak form characterizes maximal ideals by evaluation at field-valued points.

• Maximal elementBroader topic: It is maximality in the partially ordered set of proper ideals ordered by inclusion.

• Local ringRelated: A local ring is defined by having exactly one maximal ideal.

• Nakayama's lemmaRelated: Multiplication by this ideal is the operation whose fixed finite modules Nakayama’s lemma constrains.

• Boolean prime ideal theoremCompared with: In Boolean algebras, maximal ideals are prime; the theorem guarantees such ideals exist.

• Krull's principal ideal theoremCompared with: A maximal ideal containing one element need not have height at most one.

• Weierstrass preparation theoremRelated: Vanishing coefficients and germs at the origin are organized by the maximal ideal.

• Krull's theoremBroader topic: This is precisely the object whose existence Krull's theorem guarantees.

• Gelfand–Mazur theoremRelated: Quotients by maximal ideals produce complex Banach division algebras.

• Teichmüller–Tukey lemmaBroader topic: The lemma can extend a proper ideal to a maximal one because properness is preserved by chain unions.