Principal ideal - Revision history

Vipul: 5 revisions

2008-05-12T16:33:15Z

5 revisions

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Vipul at 17:04, 3 March 2008

2008-03-03T17:04:28Z

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			{{basicdef}}
	{{curing-ideal property}}		{{curing-ideal property}}

	{{ideal-as-a-module\|cyclic module}}		{{ideal-as-a-module\|cyclic module}}

Vipul at 16:09, 11 January 2008

2008-01-11T16:09:37Z

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	For a [[gcd domain]], it is true that given any [[finitely generated 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]].

Vipul at 16:07, 11 January 2008

2008-01-11T16:07:43Z

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	{{ideal-as-a-module\|cyclic module}}		{{ideal-as-a-module\|cyclic module}}

	==Definition ~~for commutative rings~~==		==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>.

	==~~Definition for noncommutative rings~~==		==Relation with other properties==

	~~{{fillin}}~~		===Weaker properties===

	[[~~Category: Properties~~ of ideals ~~in commutative rings~~]]		* [[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.

Vipul at 16:02, 11 January 2008

2008-01-11T16:02:49Z

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			{{curing-ideal property}}

			{{ideal-as-a-module\|cyclic module}}

	==Definition for commutative rings==		==Definition for commutative rings==

Vipul: Started the page

2007-01-09T04:36:02Z

Started the page

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==Definition for commutative rings==

===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 <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>.

==Definition for noncommutative rings==

{{fillin}}

[[Category: Properties of ideals in commutative rings]]