# Finitely generated ideal

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 ideals in commutative unital rings depends only on the ideal, viewed abstractly as a module over the commutative unital ring. The corresponding module property that the ideal must satisfy is:finitely generated module

View other such properties

## Contents

## Definition

### Symbol-free definition

An ideal in a commutative unital ring is said to be **finitely generated** if it has a finite generating set, that is, if there is a finite set such that it is the smallest ideal containing that finite set.

### Definition with symbols

An ideal in a commutative unital ring is said to be **finitely generated** if there is a finite set such that .

## Relation with other properties

### Stronger properties

## Metaproperties

### Trimness

*This property of ideals in commutative unital rings is trim: it is satisfied by the zero ideal and by the whole ring, as an ideal of itself*Template:Finite-sum-closed ideal property

### 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*

In general, an intersection of finitely generated ideals need not be finitely generated. However, for Noetherian rings, where every ideal is finitely generated, an intersection of finitely generated ideals is certainly finitely generated.