Maximal ideal

In mathematics, more specifically in ring theory, a maximal ideal is a two-sided ideal that is maximal (with respect to set inclusion) amongst all proper ideals. In other words, I is a maximal ideal of a ringR if there are no other two-sided ideals contained between I and R.

Maximal ideals are important because the quotients of rings by maximal ideals are simple rings, and in the special case of commutative rings they are also fields. The set of maximal ideals of a commutative ring R is known as the maximal spectrum of R and is variously denoted m-Spec R, Specm R, MaxSpec R, Max R, or Spm R. When equipped with the Zariski topology, it is the subspace of the closed points of the prime spectrum Spec R, the set of the prime ideals equipped with Zariski topology.

In the theory of noncommutative rings a maximal right ideal is defined analogously as being a maximal element in the poset of proper right ideals, and similarly, a maximal left ideal is defined to be a maximal element of the poset of proper left ideals. Since a one-sided maximal ideal A is not necessarily two-sided, the quotient R/A is not necessarily a ring, but it is a simple module over R. If R has a unique maximal right ideal, then R is known as a local ring, and the maximal right ideal is also the unique maximal left and unique maximal two-sided ideal of the ring, and is in fact the Jacobson radical J(R).

It is possible for a ring to have a unique maximal two-sided ideal and yet lack unique maximal one-sided ideals: for example, in the ring of 2 by 2 square matrices over a field, the zero ideal is a maximal two-sided ideal, but there are many maximal right ideals.