Unique factorization domain - Revision history

Vipul: /* Metaproperties */

2009-02-09T03:43:10Z

Metaproperties

← Older revision		Revision as of 03:43, 9 February 2009
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	{{further\|[[Unique factorization is localization-closed]]}}		{{further\|[[Unique factorization is localization-closed]]}}

	{{prime-quotient-closed idp}}		{{not prime-quotient-closed idp}}

	The quotient of a unique factorization domain by a prime ideal ~~is again~~ a unique factorization domain. ~~The irreducibles in~~ the ~~new~~ unique factorization domain ~~are simply the irreducibles outside~~ the prime ideal ~~being quotiented~~.		The quotient of a unique factorization domain by a prime ideal need ''not'' be a unique factorization domain. For instance, the ring <math>\mathbb{Z}[\sqrt{-5}]</math>, which is not a unique factorization domain, is the quotient <math>\mathbb{Z}[x]/(x^2 + 5)</math> of a unique factorization domain by a prime ideal. Similarly, the [[ring of trigonometric polynomials]] <math>\R[x,y]/(x^2 + y^2 - 1)</math> is not a unique factorization domain.

	{{further\|[[Unique factorization is prime-quotient-closed]]}}		{{further\|[[Unique factorization is not prime-quotient-closed]]}}

	===Ring of integer-valued polynomials===		===Ring of integer-valued polynomials===

	The ring of integer-valued polynomials over a unique factorization domain need not be a unique factorization domain. The simplest example is where the base ring is the [[ring of rational integers]]. {{further\|[[Ring of integer-valued polynomials over ring of rational integers is not a UFD]]}}		The ring of integer-valued polynomials over a unique factorization domain need not be a unique factorization domain. The simplest example is where the base ring is the [[ring of rational integers]]. {{further\|[[Ring of integer-valued polynomials over ring of rational integers is not a UFD]]}}

Vipul at 00:34, 6 February 2009

2009-02-06T00:34:25Z

← Older revision		Revision as of 00:34, 6 February 2009
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	===Conjunction with other properties===		===Conjunction with other properties===

	* [[Principal ideal domain]] is the conjunction with the property of being a [[Dedekind domain]].		* [[Principal ideal domain]] is the conjunction with the property of being a [[Dedekind domain]]. {{proofat\|[[Unique factorization and Dedekind iff principal ideal]]}}
			* [[Principal ideal domain]] is the conjunction with the property of being a [[Bezout domain]]. {{proofat\|[[Unique factorization and Bezout iff principal ideal]]}}

	===Stronger properties===		===Stronger properties===
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	===Incomparable properties===		===Incomparable properties===

	* [[Noetherian domain]]		* [[Noetherian domain]]: {{proofat\|[[Noetherian not implies UFD]]\|[[UFD not implies Noetherian]]}}

	==Metaproperties==		==Metaproperties==
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	{{localization-closed idp}}		{{localization-closed idp}}

	~~In fact, it is true more generally, that the [[localization at a multiplicatively closed subset]] of a unique factorization domain, is a unique factorization domain.~~ Those irreducibles which occur in the saturation of the multiplicatively closed subset that we invert, no longer remain irreducibles; the others continue to remain irreducible.		Those irreducibles which occur in the saturation of the multiplicatively closed subset that we invert, no longer remain irreducibles; the others continue to remain irreducible.

			{{further\|[[Unique factorization is localization-closed]]}}

	{{prime-quotient-closed idp}}		{{prime-quotient-closed idp}}

	The quotient of a unique factorization domain by a prime ideal is again a unique factorization domain. The irreducibles in the new unique factorization domain are simply the irreducibles outside the prime ideal being quotiented.		The quotient of a unique factorization domain by a prime ideal is again a unique factorization domain. The irreducibles in the new unique factorization domain are simply the irreducibles outside the prime ideal being quotiented.

			{{further\|[[Unique factorization is prime-quotient-closed]]}}

	===Ring of integer-valued polynomials===		===Ring of integer-valued polynomials===

	The ring of integer-valued polynomials over a unique factorization domain need not be a unique factorization domain. The simplest example is where the base ring is the [[ring of rational integers]]. {{further\|[[Ring of integer-valued polynomials over ring of rational integers is not a UFD]]}}		The ring of integer-valued polynomials over a unique factorization domain need not be a unique factorization domain. The simplest example is where the base ring is the [[ring of rational integers]]. {{further\|[[Ring of integer-valued polynomials over ring of rational integers is not a UFD]]}}

Vipul: /* Metaproperties */

2009-02-01T16:38:55Z

Metaproperties

← Older revision		Revision as of 16:38, 1 February 2009
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	{{poly-closed idp}}		{{poly-closed idp}}

	~~This is essentially what~~ [[~~Gauss's lemma~~]] ~~states.~~		{{proofat\|[[Unique factorization is polynomial-closed]]}}

	{{localization-closed idp}}		{{localization-closed idp}}

	In fact, it is true more generally, that the [[localization at a multiplicatively closed subset]] of a unique factorization domain, is a unique factorization domain. Those irreducibles which occur in the saturation of the multiplicatively closed subset that we invert, no longer remain irreducibles; the others continue to remain irreducible.		In fact, it is true more generally, that the [[localization at a multiplicatively closed subset]] of a unique factorization domain, is a unique factorization domain. Those irreducibles which occur in the saturation of the multiplicatively closed subset that we invert, no longer remain irreducibles; the others continue to remain irreducible.

			{{prime-quotient-closed idp}}

			The quotient of a unique factorization domain by a prime ideal is again a unique factorization domain. The irreducibles in the new unique factorization domain are simply the irreducibles outside the prime ideal being quotiented.

			===Ring of integer-valued polynomials===

			The ring of integer-valued polynomials over a unique factorization domain need not be a unique factorization domain. The simplest example is where the base ring is the [[ring of rational integers]]. {{further\|[[Ring of integer-valued polynomials over ring of rational integers is not a UFD]]}}

Vipul: /* Relation with other properties */

2009-01-17T17:21:44Z

Relation with other properties

← Older revision		Revision as of 17:21, 17 January 2009
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	==Relation with other properties==		==Relation with other properties==

			===Conjunction with other properties===

			* [[Principal ideal domain]] is the conjunction with the property of being a [[Dedekind domain]].

	===Stronger properties===		===Stronger properties===

	* [[Euclidean domain]]		* [[Weaker than::Euclidean domain]]
	* [[Principal ideal domain]]: {{proofofstrictimplicationat\|[[PID implies UFD]]\|[[UFD not implies PID]]}}		* [[Weaker than::Principal ideal domain]]: {{proofofstrictimplicationat\|[[PID implies UFD]]\|[[UFD not implies PID]]}}
			* [[Weaker than::Noetherian unique factorization domain]]

	===Weaker properties===		===Weaker properties===

	* [[Normal domain]]: {{proofat\|[[Unique factorization implies normal]]}}		* [[Stronger than::Normal domain]]: {{proofat\|[[Unique factorization implies normal]]}}
	* [[gcd domain]]: {{proofat\|[[UFD implies gcd]]}}		* [[Stronger than::gcd domain]]: {{proofat\|[[UFD implies gcd]]}}
	* [[Ring satisfying ACCP]]		* [[Stronger than::Ring satisfying ACCP]]

			===Incomparable properties===

			* [[Noetherian domain]]

	==Metaproperties==		==Metaproperties==

Vipul: 7 revisions

2008-05-12T16:35:01Z

7 revisions

← Older revision	Revision as of 16:35, 12 May 2008
(No difference)

Vipul: /* Metaproperties */

2008-03-15T23:15:38Z

Metaproperties

← Older revision		Revision as of 23:15, 15 March 2008
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	This is essentially what [[Gauss's lemma]] states.		This is essentially what [[Gauss's lemma]] states.

			{{localization-closed idp}}

			In fact, it is true more generally, that the [[localization at a multiplicatively closed subset]] of a unique factorization domain, is a unique factorization domain. Those irreducibles which occur in the saturation of the multiplicatively closed subset that we invert, no longer remain irreducibles; the others continue to remain irreducible.

Vipul: /* Weaker properties */

2008-01-08T01:19:34Z

Weaker properties

← Older revision		Revision as of 01:19, 8 January 2008
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	* [[Normal domain]]: {{proofat\|[[Unique factorization implies normal]]}}		* [[Normal domain]]: {{proofat\|[[Unique factorization implies normal]]}}
	* [[gcd domain]]: {{proofat\|[[UFD implies gcd]]}}		* [[gcd domain]]: {{proofat\|[[UFD implies gcd]]}}
			* [[Ring satisfying ACCP]]

	==Metaproperties==		==Metaproperties==

Vipul: /* Definition */

2008-01-07T23:52:19Z

Definition

← Older revision		Revision as of 23:52, 7 January 2008
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	===Symbol-free definition===		===Symbol-free definition===

	An [[integral domain]] is termed a '''unique factorization domain''' if every element can be expressed as a product of finite length of [[irreducible element]]s (possibly with multiplicity) in a manner that is unique upto the ordering of the elements.		An [[integral domain]] is termed a '''unique factorization domain''' or '''factorial domain''' if every element can be expressed as a product of finite length of [[irreducible element]]s (possibly with multiplicity) in a manner that is unique upto the ordering of the elements.

	===Definition with symbols===		===Definition with symbols===

Vipul: /* Stronger properties */

2007-12-16T23:04:44Z

Stronger properties

← Older revision		Revision as of 23:04, 16 December 2007
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	* [[Euclidean domain]]		* [[Euclidean domain]]
	* [[Principal ideal domain]]		* [[Principal ideal domain]]: {{proofofstrictimplicationat\|[[PID implies UFD]]\|[[UFD not implies PID]]}}

	===Weaker properties===		===Weaker properties===

Vipul: /* Weaker properties */

2007-08-08T08:49:59Z

Weaker properties

← Older revision		Revision as of 08:49, 8 August 2007
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	===Weaker properties===		===Weaker properties===

	* [[Normal domain]]: {{proofat\|[[~~UFD~~ implies normal]]}}		* [[Normal domain]]: {{proofat\|[[Unique factorization implies normal]]}}
	* [[gcd domain]]: {{proofat\|[[UFD implies gcd]]}}		* [[gcd domain]]: {{proofat\|[[UFD implies gcd]]}}