Maximal ideal

A maximal ideal is a proper ideal of a ring that is contained in no larger proper ideal. Equivalently, its quotient ring is a field when the ring is commutative with identity.

Ring: A set with addition and multiplication satisfying the axioms of a ring. Maximal ideals are defined inside rings, whose conventions affect the theorem’s scope.
Prime ideal: A proper ideal whose quotient ring is an integral domain. Prime ideals require a domain quotient, while maximal ideals require a field quotient.
Hilbert's Nullstellensatz: A theorem relating maximal ideals of polynomial rings over algebraically closed fields to points of affine space. It identifies maximal ideals in polynomial rings with algebraic points under its hypotheses.
Radical ideal: An ideal I such that xⁿ in I implies x in I for every ring element x and positive integer n. Every maximal ideal is radical in a commutative ring, but radical ideals need not be maximal.
Ideal: An additive subgroup of a ring closed under multiplication by ring elements. Maximality compares an ideal with every larger ideal in the same ring.
Maximal ideal theorem: The theorem that every proper ideal in a unital ring is contained in a maximal ideal. It guarantees maximal ideals by extending a proper ideal as far as possible.
Maximal spectrum: The set of maximal ideals of a commutative ring, often equipped with the Zariski topology. It organizes maximal ideals as the ring’s space of closed-point candidates.
Primary ideal: An ideal I such that xy in I and x not in I imply some power of y lies in I. Primary ideals generalize prime-like behavior without demanding maximal containment.
Proper ideal: An ideal that is not equal to its ambient ring. Only proper ideals qualify for maximality.
Zorn's lemma: A set-theoretic principle stating that a partially ordered set with upper bounds for chains has a maximal element. Its application to proper ideals proves the maximal ideal theorem.

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