Principal ideal: Difference between revisions
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==Definition | {{basicdef}} | ||
{{curing-ideal property}} | |||
{{ideal-as-a-module|cyclic module}} | |||
==Definition== | |||
===Symbol-free definition=== | ===Symbol-free definition=== | ||
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An ideal <math>I</math> in a ring <math>R</math> is termed a '''principal ideal''' if there exists an <math>x</math> in <math>R</math> such that <math>I = Rx</math>. | An ideal <math>I</math> in a ring <math>R</math> is termed a '''principal ideal''' if there exists an <math>x</math> in <math>R</math> such that <math>I = Rx</math>. | ||
== | ==Relation with other properties== | ||
===Weaker properties=== | |||
* [[Finitely generated ideal]] | |||
==Metaproperties== | |||
{{trim ideal property}} | |||
The whole ring, as well as the zero ideal, are principal ideals. | |||
{{not intersection-closed ideal property}} | |||
An intersection of principal ideals need not be a principal ideal. However, for a [[unique factorization domain]], it ''is'' true that an arbitrary intersection of principal ideals is principal. Thus, given any ideal, there exists a ''smallest'' principal ideal containing it. | |||
For a [[gcd domain]], it is true that given any [[finitely generated ideal]], there exists a ''smallest'' principal ideal containing it. | |||
{{ | {{not sum-closed ideal property}} | ||
[[ | The property of being a principal ideal is not closed under taking finite, or arbitrary sums, of ideals. A finite sum of principal ideals is a [[finitely generated ideal]], and every finitely generated ideal is principal iff the ring is a [[Bezout ring]]. Any ideal can be expressed as an arbitrary sum of principal ideals, and so an arbitrary sum of principal ideals is a principal ideal iff the ring is a [[principal ideal ring]]. | ||
Latest revision as of 16:33, 12 May 2008
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 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: cyclic module
View other such properties
Definition
Symbol-free definition
An ideal in a commutative unital ring is termed a principal ideal if it is the ideal generated by a single element of the ring.
Definition with symbols
An ideal in a ring is termed a principal ideal if there exists an in such that .
Relation with other properties
Weaker 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
The whole ring, as well as the zero ideal, are principal ideals.
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
An intersection of principal ideals need not be a principal ideal. However, for a unique factorization domain, it is true that an arbitrary intersection of principal ideals is principal. Thus, given any ideal, there exists a smallest principal ideal containing it.
For a gcd domain, it is true that given any finitely generated ideal, there exists a smallest principal ideal containing it.
Template:Not sum-closed ideal property
The property of being a principal ideal is not closed under taking finite, or arbitrary sums, of ideals. A finite sum of principal ideals is a finitely generated ideal, and every finitely generated ideal is principal iff the ring is a Bezout ring. Any ideal can be expressed as an arbitrary sum of principal ideals, and so an arbitrary sum of principal ideals is a principal ideal iff the ring is a principal ideal ring.