# 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