Maximal ideal

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.^[1][2] In other words, I is a maximal ideal of a ring R if I is an ideal of R, I ≠ R, and whenever J is another ideal containing I as a subset, then either J = I or J = R. So there are no ideals "in 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 in the poset of right ideals, and similarly, a maximal left ideal is defined. Since a one sided maximal ideal A is not necessarily twosided, 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 twosided 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.

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.

Equivalently, I is a maximal ideal of R if I ≠ R and for all ideals J with I ⊆ J, either J = I or J = R.

Examples

• In the ring Z of integers the maximal ideals are the principal ideals generated by a prime number.
• More generally, all nonzero prime ideals are maximal in a principal ideal domain.
• The maximal ideals of the polynomial ring K[x₁,...,x_n] over an algebraically closed field K are the ideal of the form (x₁ − a₁,...,x_n − a_n). This result is known as the weak nullstellensatz.

Properties

• If R is a unital commutative ring with an ideal m, then k = R/m is a field if and only if m is a maximal ideal. In that case, R/m is known as the residue field. This fact can fail 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.

• If L is a maximal left ideal, then R/L is a simple left R module. Conversely in rings with unity, any simple left R module arises this way. Incidentally this shows that a collection of representatives of simple left R modules is actually a set since it can be put into correspondence with part of the set of maximal left ideals of R.

• Krull's theorem (1929): Every ring with a multiplicative identity has a maximal ideal. The result is also true if "ideal" is replaced with "right ideal" or "left ideal". More generally, it is true that every nonzero finitely generated module has a maximal submodule. Suppose I is an ideal which is not R (respectively, A is a right ideal which is not R). Then R/I is a ring with unity, (respectively, R/A is a finitely generated module), and so the above theorems can be applied to the quotient to conclude that there is a maximal ideal (respectively maximal right ideal) of R containing I (respectively, A).

• Krull's theorem can fail for rings without unity. A radical ring, i.e. a ring in which the Jacobson radical is the entire ring, has no simple modules and hence has no maximal right or left ideals.

• In a commutative ring with unity, every maximal ideal is a prime ideal. The converse is not always true: for example, in any nonfield integral domain the zero ideal is a prime ideal which is not maximal. Commutative rings in which prime ideals are maximal are known as zero-dimensional rings, where the dimension used is the Krull dimension.

References

1. ^ Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). John Wiley & Sons. ISBN 0-471-43334-9. 
2. ^ Lang, Serge (2002). Algebra. Graduate Texts in Mathematics. Springer. ISBN 0-387-95385-X.