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. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... In mathematics, ring theory is the study of rings, algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers. ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ...

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. In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. ... In abstract algebra, a simple ring is a non-zero ring that has no ideal besides the zero ideal and itself. ... In mathematics, an associative algebra is unital if it contains a multiplicative identity element, i. ... In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, more particularly in abstract algebra, local rings are certain rings that are comparatively simple, and serve to describe the local behavior of functions defined on varieties or manifolds. ...

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.

Examples

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

In Ring theory, a branch of abstract algebra, a principal ideal is an ideal I in a ring R that is generated by a single element a of R. More specifically: a left principal ideal of R is a subset of R of the form Ra := {ra : r in R...

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.

• 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, is a maximal ideal in , but is not a field.