Maximal ideal

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In mathematics, more specifically in ring theory a maximal ideal is an ideal which is maximal (with respect to set inclusion) amongst all proper ideals, i.e. which is not contained in any other proper 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.

[edit] Definition

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

[edit] Examples

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

[edit] Properties

In a commutative ring with unity, 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.

In a unital commutative ring, an ideal is maximal if and only if its factor ring is a field. This fails in non-unital rings. For example, $4\mathbb{Z}$ is a maximal ideal in $2\mathbb{Z}$ , but $2\mathbb{Z}/4\mathbb{Z}$ is not a field.