Maximal ideal

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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 R on R/M makes R/M into a simple R-module.

Definition with symbols

An ideal M in a commutative ring R is termed maximal if it satisfies the following equivalent conditions:

  • For any ideal J such that MJR, J is equal either to M or to R.
  • The quotient ring R/M is a field.

Relation with other properties

Weaker properties

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

External links

Definition links