Maximal ideal

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

Maximal ideals are important because the quotient rings of maximal ideals are simple rings, and in the special case of unital commutative rings they are also fields.

In noncommutative ring theory, 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 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 ideal, but there are many maximal right ideals.

There are other equivalent ways of expressing the definition of maximal one-sided and maximal two-sided ideals. Given a ring R and a proper ideal I of R (that is I ≠ R), I is a maximal ideal of R if any of the following equivalent conditions hold:

There is an analogous list for one-sided ideals, for which only the right-hand versions will be given. For a right ideal A of a ring R, the following conditions are equivalent to A being a maximal right ideal of R:

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