Maximal ideal
From Commalg
This article is about a basic definition in commutative algebra. View a complete list of basic definitions in commutative algebra
This article defines a property of an ideal in a commutative unital ring |View other properties of ideals in commutative unital rings
This property of an ideal in a ring is equivalent to the property of the quotient ring being a/an: field | View other quotient-determined properties of ideals in commutative unital rings
Definition
Symbol-free definition
An ideal in a commutative unital ring (or more generally, in any commutative ring) is termed maximal if it is proper (not the whole ring) and it satisfies the following equivalent conditions:
- There is no ideal of the ring properly in between this ideal and the whole ring
- The quotient of the ring by this ideal is a field
- The natural action of
on 
makes into a simple -module.
Definition with symbols
An ideal 
in a commutative ring is termed maximal if it satisfies the following equivalent conditions:
- For any ideal
such that ≤ ≤ , is equal either to or to . - The quotient ring is a field.
Relation with other properties
Weaker properties
- Prime ideal: For proof of the implication, refer Maximal implies prime and for proof of its strictness (i.e. the reverse implication being false) refer prime not implies maximal
- Radical ideal
- Intersection of maximal ideals
Metaproperties
Intersection-closedness
This property of ideals in commutative unital rings is not closed under taking arbitrary intersections; in other words, an arbitrary intersection of ideals with this property need not have this property
Template:Not intringcondn ideal
Effect of property operators
The intersection-closure
Applying the intersection-closure to this property gives: intersection of maximal ideals
