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. Rings which contain only one maximal ideal are called local rings.

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, this holds for any 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 m is a maximal ideal and R is commutative ring, then k = R/m is a field, known as the residue field

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 ring with a multiplicative identity 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.

Every nonzero ring has a maximal ideal.

In a unital ring, every proper ideal is contained in a maximal ideal.

References

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

[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.

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