# Intersection of maximal ideals

From Commalg

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: semisimple ring | View other quotient-determined properties of ideals in commutative unital rings

## Contents

## Definition

### Symbol-free definition

An ideal in a commutative unital ring is termed an **intersection of maximal ideals** if it can be expressed as an intersection of maximal ideals (this is really a tautological definition).

Note that although a maximal ideal is assumed to be proper, an *intersection of maximal ideals* could be the whole ring, i.e. we allow the empty intersection.

## Relation with other properties

### Stronger properties

### Weaker properties

- Radical ideal:
*For full proof, refer: Intersection of maximal ideals implies radical*

### Related ring properties

A ring in which every radical ideal is an intersection of maximal ideals is termed a Jacobson ring (or sometimes a Hilbert ring).

## Metaproperties

### Intersection-closedness

*This property of ideals in commutative unital rings is intersection-closed: an arbitrary intersection of ideals with this property, also has this property*