Finitely generated ideal: Difference between revisions
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==Definition | {{curing-ideal property}} | ||
{{ideal-as-a-module|finitely generated module}} | |||
==Definition== | |||
===Symbol-free definition=== | ===Symbol-free definition=== | ||
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An [[ideal]] <math>I</math> in a [[commutative unital ring]] <math>R</math> is said to be '''finitely generated''' if there is a finite set <math>x_1, x_2, ..., x_n</math> such that <math>I = Rx_1 + Rx_2 + ... + Rx_n</math>. | An [[ideal]] <math>I</math> in a [[commutative unital ring]] <math>R</math> is said to be '''finitely generated''' if there is a finite set <math>x_1, x_2, ..., x_n</math> such that <math>I = Rx_1 + Rx_2 + ... + Rx_n</math>. | ||
== | ==Relation with other properties== | ||
===Stronger properties=== | |||
== | * [[Principal ideal]] | ||
==Metaproperties== | |||
{{trim ideal property}} | |||
{{finite-sum-closed ideal property}} | |||
{{ | {{not intersection-closed ideal property}} | ||
[[ | In general, an intersection of finitely generated ideals need not be finitely generated. However, for [[Noetherian ring]]s, where every ideal is finitely generated, an intersection of finitely generated ideals is certainly finitely generated. | ||
Revision as of 16:20, 11 January 2008
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
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.