In mathematics, more specifically in ring theory a maximal ideal is a special kind of ideal which is in some sense maximal, that is not contained in any other non-trivial ideal of the ring.

Maximal ideals are important because the quotient rings of maximal ideals are simple rings and in the special case of unital commutative rings even fields. Rings which contain only one maximal ideal are called local rings

Definition

Given a ring R and a proper ideal I of R (that is I ≠ R), I is called maximal ideal in R if there exists no other proper ideal J of R so that I ⊂ J.

An ideal S of a ring such that S≠ R is called a maximal ideal of R if there exists no proper ideals of R contaning S.

Examples

• In the ring Z of integers the maximal ideals are the principal ideals generated by a prime number.

Properties

• Every maximal ideal is a prime ideal. Maximal ideals can be directly characterized to be those ideals which are subsets of only two ideals: the improper ideal and the maximal ideal itself.

• Krull's theorem (1929): Every commutative ring with 1 has a maximal ideal.

• In a lattice diagram, maximal ideals are always directly joined to the biggest containing ring, as follows from the prime property.

Ring theory