Principal ideal domain - Revision history

Vipul: /* Definition */

2014-01-29T20:33:08Z

Definition

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	* For (1) implies (3), refer [[principal ideal domain admits multiplicative Dedekind-Hasse norm]].		* For (1) implies (3), refer [[principal ideal domain admits multiplicative Dedekind-Hasse norm]].
	* For (3) implies (1), refer [[Dedekind-Hasse norm implies principal ideal ring]].		* For (3) implies (1), refer [[Dedekind-Hasse norm implies principal ideal ring]].

			==Examples==

			Since [[Euclidean implies PID]], all the [[Euclidean domain#Examples\|typical examples of Eulidean domains]] are also examples of principal ideal domains. There are some PIDs that are not Euclidean domains -- in particular, the domain <math>\mathbb{Z}\left[\frac{1 + \sqrt{-19}}{2}\right]</math> is not a Euclidean domain because it does not have a universal side divisor.

	==Relation with other properties==		==Relation with other properties==

Vipul: /* Definition */

2013-07-20T15:55:22Z

Definition

← Older revision		Revision as of 15:55, 20 July 2013
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	==Definition==		==Definition==

	~~Below are some '''equivalent''' definitions of~~ '''principal ideal domain''' (also known as '''PID''') if it is an [[integral domain]] and a [[principal ideal ring]]. Explicitly, it must satisfy the following '''equivalent''' conditions:		A [[commutative unital ring]] is termed a '''principal ideal domain''' (also known as '''PID''') if it is an [[integral domain]] and a [[principal ideal ring]]. Explicitly, it must satisfy the following '''equivalent''' conditions:

	{\| class="sortable" border="1"		{\| class="sortable" border="1"

Vipul: /* Weaker properties */

2013-07-18T16:04:32Z

Weaker properties

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	\| [[Stronger than::Noetherian ring]] \|\| \|\| \|\| \|\| {{intermediate notions short\|Noetherian ring\|principal ideal domain}}		\| [[Stronger than::Noetherian ring]] \|\| \|\| \|\| \|\| {{intermediate notions short\|Noetherian ring\|principal ideal domain}}
	\|-		\|-
	\| [[Stronger than::~~Unique~~ factorization domain]] \|\| [[integral domain]] with unique factorization \|\| [[PID implies UFD]]\|\|[[UFD not implies PID]] \|\| {{intermediate notions short\|unique factorization domain\|principal ideal domain}}		\| [[Stronger than::unique factorization domain]] \|\| [[integral domain]] with unique factorization \|\| [[PID implies UFD]]\|\|[[UFD not implies PID]] \|\| {{intermediate notions short\|unique factorization domain\|principal ideal domain}}
	\|-		\|-
	\| [[Stronger than::~~Normal~~ domain]] \|\| [[integral domain]] that is integrally closed in its field of fractions \|\| (via UFD) \|\| (via UFD) \|\| {{intermediate notions short\|normal domain\|principal ideal domain}}		\| [[Stronger than::normal domain]] \|\| [[integral domain]] that is integrally closed in its field of fractions \|\| (via UFD) \|\| (via UFD) \|\| {{intermediate notions short\|normal domain\|principal ideal domain}}
	\|-		\|-
	\| [[Stronger than::Noetherian normal domain]] \|\| [[Noetherian ring]] and [[normal domain]] \|\| \|\| \|\| {{intermediate notions short\|Noetherian normal domain\|principal ideal domain}}		\| [[Stronger than::Noetherian normal domain]] \|\| [[Noetherian ring]] and [[normal domain]] \|\| \|\| \|\| {{intermediate notions short\|Noetherian normal domain\|principal ideal domain}}
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	\| [[Stronger than::One-dimensional Noetherian domain]] \|\| [[one-dimensional ring\|one-dimensional]] and [[Noetherian domain\|Noetherian]] \|\| \|\| \|\| {{intermediate notions short\|one-dimensional Noetherian domain\|principal ideal domain}}		\| [[Stronger than::One-dimensional Noetherian domain]] \|\| [[one-dimensional ring\|one-dimensional]] and [[Noetherian domain\|Noetherian]] \|\| \|\| \|\| {{intermediate notions short\|one-dimensional Noetherian domain\|principal ideal domain}}
	\|-		\|-
	\| [[Stronger than::~~Elementary~~ divisor domain]] \|\| \|\| \|\| \|\| {{intermediate notions short\|elementary divisor domain\|principal ideal domain}}		\| [[Stronger than::elementary divisor domain]] \|\| \|\| \|\| \|\| {{intermediate notions short\|elementary divisor domain\|principal ideal domain}}
	\|-		\|-
	\| [[Stronger than::gcd domain]] \|\| \|\| \|\| \|\| {{intermediate notions short\|gcd domain\|principal ideal domain}}		\| [[Stronger than::gcd domain]] \|\| \|\| \|\| \|\| {{intermediate notions short\|gcd domain\|principal ideal domain}}
	\|-		\|-
	\| [[Stronger than::~~Principal~~ ideal ring]] \|\| every ideal is [[principal ideal\|principal]], not necessarily a [[integral domain\|domain]] \|\| \|\| \|\| {{intermediate notions short\|principal ideal ring\|principal ideal domain}}		\| [[Stronger than::principal ideal ring]] \|\| every ideal is [[principal ideal\|principal]], not necessarily a [[integral domain\|domain]] \|\| \|\| \|\| {{intermediate notions short\|principal ideal ring\|principal ideal domain}}
			\|-
			\| [[Stronger than::catenary ring]] \|\| any two saturated chains of primes with the same starting and ending point have the same length \|\| \|\| \|\| {{intermediate notions short\|catenary ring\|principal ideal domain}}
	\|}		\|}

Vipul: /* Metaproperties */

2013-07-18T03:33:26Z

Metaproperties

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	==Metaproperties==		==Metaproperties==

	{~~{not poly~~-~~closed idp}}~~		{\| class="sortable" border="1"
			! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols
	~~The~~ polynomial ring over a ~~PID need not be~~ a PID~~. Two examples are~~ the ~~polynomial~~ ring ~~over the integers, and~~ the polynomial ring ~~in two variables over~~ a ~~field~~. In fact, ~~the polynomial ring over a ring~~ is a ~~PID iff that ring~~ is a field. {{~~proofat\|[~~[~~Polynomial ring over a ring is a PID iff the ring is a field~~]]}}		\|-
			\| [[dissatisfies metaproperty::polynomial-closed property of commutative unital rings]] \|\| No \|\| follows from [[Polynomial ring over a ring is a PID iff the ring is a field]] \|\| It is possible to have a principal ideal domain <math>R</math> such that the polynomial ring <math>R[x]</math> is not a principal ideal domain. In fact, <math>R[x]</math> is a principal ideal domain if and only if <math>R</math> is a field, so for instance <math>R = \mathbb{Z}</math> or <math>R = \mathbb{R}[x]</math> give counterexamples.
			\|}

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

Vipul: /* Relation with other properties */

2013-07-18T03:29:22Z

Relation with other properties

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	===Expression as a conjunction of other properties===		===Expression as a conjunction of other properties===

	* A [[principal ideal ring]] ~~that is also an~~ [[integral domain]].		{\| class="sortable" border="1"
	* A [[conjunction involving::Noetherian domain]] ~~that is also a~~ [[conjunction involving::Bezout domain]] ~~(equivalently, an [[integral domain]] that is both a [[conjunction involving::Noetherian ring]] and a [[conjunction involving::Bezout ring]]). {{proofat~~\|[[Noetherian and Bezout iff principal ideal]]}}		! First component of conjunction !! Second component of conjunction !! Proof
	* A [[unique factorization domain]] ~~that is also a~~ [[Bezout domain]]~~. {{proofat~~\|[[Unique factorization and Bezout iff principal ideal]]}}		\|-
	* A [[conjunction involving::unique factorization domain]] ~~that is also a~~ [[conjunction involving::Dedekind domain]]~~. More generally, a~~ unique factorization domain ~~that is also a~~ [[conjunction involving::one-dimensional domain]]~~.{{proofat~~\|[[Unique factorization and one-dimensional iff principal ideal]]}}		\| [[principal ideal ring]] \|\| [[integral domain]] \|\| by definition
			\|-
			\| [[conjunction involving::Noetherian domain]] \|\| [[conjunction involving::Bezout domain]] \|\| [[Noetherian and Bezout iff principal ideal]]
			\|-
			\| [[conjunction involving::unique factorization domain]] \|\| [[Bezout domain]] \|\| [[Unique factorization and Bezout iff principal ideal]]
			\|-
			\| [[conjunction involving::unique factorization domain]] \|\| [[conjunction involving::Dedekind domain]] \|\| (see next)
			\|-
			\| [[unique factorization domain]] \|\| [[conjunction involving::one-dimensional domain]] \|\| [[Unique factorization and one-dimensional iff principal ideal]]
			\|}

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

Vipul: /* Definition */

2013-07-18T03:22:47Z

Definition

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	==Definition==		==Definition==

	~~===Symbol-free definition===~~		Below are some '''equivalent''' definitions of '''principal ideal domain''' (also known as '''PID''') if it is an [[integral domain]] and a [[principal ideal ring]]. Explicitly, it must satisfy the following '''equivalent''' conditions:

	~~An [[integral~~ domain]] is termed a ~~'''PID''' or '''Principal Ideal Domain'''~~ if ~~it satisfies the following equivalent conditions:~~		{\| class="sortable" border="1"
			! No. !! Shorthand !! A commutative unital ring is termed a principal ideal domain if ... !! A commutative unital ring <math>R</math> is termed a principal ideal domain if ...
	~~# Every~~ [[ideal]] in it is [[defining ingredient::principal ideal\|principal]], ~~viz~~., it is a [[defining ingredient::~~principal ideal ring]]~~		\|-
	~~# Every [[~~prime ideal]] in it is [[principal ideal\|principal]]		\| 1 \|\| every ideal principal \|\| it is an [[integral domain]] and every [[ideal]] in it is [[defining ingredient::principal ideal\|principal]] \|\| <math>R</math> is an integral domain (i.e., <math>ab = 0</math> implies <math>a = 0</math> or <math>b = 0</math>) and for every ideal <math>I \subseteq R</math>, there exists <math>x \in I</math> such that <math>I = (x)</matH>, i.e., <math>I = Rx</math>.
	~~# It~~ admits a [[defining ingredient::Dedekind-Hasse norm]]		\|-
			\| 2 \|\| every prime ideal principal \|\| it is an [[integral domain]] and every [[defining ingredient::prime ideal]] in it is [[principal ideal\|principal]] \|\| <math>R</math> is an integral domain (i.e., <math>ab = 0</math> implies <math>a = 0</math> or <math>b = 0</math>) and for every prime ideal <math>P \subseteq R</math>, there exists <math>x \in P</math> such that <math>P = (x)</matH>, i.e., <math>P = Rx</math>.
			\|-
			\| 3 \|\| Dedekind-Hasse norm \|\| it admits a [[defining ingredient::Dedekind-Hasse norm]] \|\| there is a Dedekind-Hasse norm on <math>R</math>.
			\|}

	===Equivalence of definitions===		===Equivalence of definitions===

Vipul at 01:06, 4 July 2012

2012-07-04T01:06:24Z

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

	{\| class="~~wikitable~~" border="1"		{\| class="sortable" border="1"
	! ~~property~~ !! ~~quick description~~ !! ~~proof~~ of implication !! ~~proof~~ of strictness (reverse implication failure) !! ~~intermediate~~ notions		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
	\|-		\|-
	\| [[Weaker than::Euclidean domain]] \|\| has a [[Euclidean norm]] \|\| [[Euclidean implies PID]] \|\| [[PID not implies Euclidean]] \|\| {{intermediate notions short\|principal ideal domain\|Euclidean domain}}		\| [[Weaker than::Euclidean domain]] \|\| has a [[Euclidean norm]] \|\| [[Euclidean implies PID]] \|\| [[PID not implies Euclidean]] \|\| {{intermediate notions short\|principal ideal domain\|Euclidean domain}}
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	===Weaker properties===		===Weaker properties===

	{\| class="~~wikitable~~" border="1"		{\| class="sortable" border="1"
	! ~~property~~ !! ~~quick description~~ !! ~~proof~~ of implication !! ~~proof~~ of strictness (reverse implication failure) !! ~~intermediate~~ notions		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
	\|-		\|-
	\| [[Stronger than::Dedekind domain]] \|\| \|\| [[PID implies Dedekind]] \|\|[[Dedekind not implies PID]] \|\| {{intermediate notions short\|Dedekind domain\|principal ideal domain}}		\| [[Stronger than::Dedekind domain]] \|\| \|\| [[PID implies Dedekind]] \|\|[[Dedekind not implies PID]] \|\| {{intermediate notions short\|Dedekind domain\|principal ideal domain}}

Vipul: /* Weaker properties */

2009-10-26T21:33:29Z

Weaker properties

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	\| [[Stronger than::One-dimensional domain]]\|\| [[integral domain]] in which every nonzero [[prime ideal]] is [[maximal ideal\|maximal]] \|\| [[PID implies one-dimensional]]\|\|[[One-dimensional domain not implies PID]] \|\| {{intermediate notions short\|one-dimensional domain\|principal ideal domain}}		\| [[Stronger than::One-dimensional domain]]\|\| [[integral domain]] in which every nonzero [[prime ideal]] is [[maximal ideal\|maximal]] \|\| [[PID implies one-dimensional]]\|\|[[One-dimensional domain not implies PID]] \|\| {{intermediate notions short\|one-dimensional domain\|principal ideal domain}}
	\|-		\|-
	\| [[Stronger than::One-dimensional Noetherian domain]] \|\| [[one-dimensional ring\|one-dimensional]] and [[Noetherian domain\|Noetherian]] \|\| \|\| \|\| \|\| {{intermediate notions short\|one-dimensional Noetherian domain\|principal ideal domain}}		\| [[Stronger than::One-dimensional Noetherian domain]] \|\| [[one-dimensional ring\|one-dimensional]] and [[Noetherian domain\|Noetherian]] \|\| \|\| \|\| {{intermediate notions short\|one-dimensional Noetherian domain\|principal ideal domain}}
	\|-		\|-
	\| [[Stronger than::Elementary divisor domain]] \|\| \|\| \|\| \|\| {{intermediate notions short\|elementary divisor domain\|principal ideal domain}}		\| [[Stronger than::Elementary divisor domain]] \|\| \|\| \|\| \|\| {{intermediate notions short\|elementary divisor domain\|principal ideal domain}}

Vipul: /* Relation with other properties */

2009-10-26T21:32:22Z

Relation with other properties

← Older revision		Revision as of 21:32, 26 October 2009
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	\| [[Stronger than::Principal ideal ring]] \|\| every ideal is [[principal ideal\|principal]], not necessarily a [[integral domain\|domain]] \|\| \|\| \|\| {{intermediate notions short\|principal ideal ring\|principal ideal domain}}		\| [[Stronger than::Principal ideal ring]] \|\| every ideal is [[principal ideal\|principal]], not necessarily a [[integral domain\|domain]] \|\| \|\| \|\| {{intermediate notions short\|principal ideal ring\|principal ideal domain}}
	\|}		\|}
	~~===Conjunction expressions===~~

	~~A ring is a principal ideal domain iff it is:~~

	* A [[principal ideal ring]] and an [[integral domain]]: This is a tautological statement
	* A [[Noetherian ring]] and a [[Bezout domain]]: {{further\|[[Noetherian and Bezout iff principal ideal]]}}
	* A [[unique factorization domain]] and a [[Dedekind domain]]: {{further\|[[Unique factorization and Dedekind iff principal ideal]]}}
	* A [[unique factorization domain]] and a [[Bezout domain]]: {{further\|[[Unique factorization and Bezout iff principal ideal]]}}

	==Metaproperties==		==Metaproperties==

Vipul: /* Relation with other properties */

2009-10-26T21:31:42Z

Relation with other properties

← Older revision		Revision as of 21:31, 26 October 2009
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	===Stronger properties===		===Stronger properties===

	* [[Weaker than::Euclidean domain]]~~: {{proofofstrictimplicationat~~\|[[Euclidean implies PID]]\|[[PID not implies Euclidean]]}}		{\| class="wikitable" border="1"
	* [[Weaker than::Polynomial ring over a field]]		! property !! quick description !! proof of implication !! proof of strictness (reverse implication failure) !! intermediate notions
			\|-
			\| [[Weaker than::Euclidean domain]] \|\| has a [[Euclidean norm]] \|\| [[Euclidean implies PID]] \|\| [[PID not implies Euclidean]] \|\| {{intermediate notions short\|principal ideal domain\|Euclidean domain}}
			\|-
			\| [[Weaker than::Polynomial ring over a field]] \|\| <math>k[x]</math> for a [[field]] <math>k</math> \|\| (via Euclidean domain) \|\| (via Euclidean domain) \|\| {{intermediate notions short\|principal ideal domain\|polynomial ring over a field}}
			\|}

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

	* [[Stronger than::Dedekind domain]]~~: {{proofofstrictimplicationat~~\|[[PID implies Dedekind]]\|[[Dedekind not implies PID]]}}		{\| class="wikitable" border="1"
	* [[Stronger than::Bezout domain]]~~: {{proofofstrictimplicationat~~\|[[PID implies Bezout]]\|[[Bezout not implies PID]]}}		! property !! quick description !! proof of implication !! proof of strictness (reverse implication failure) !! intermediate notions
	* [[Stronger than::Noetherian domain]]		\|-
	* [[Stronger than::Unique factorization domain]]~~: {{proofofstrictimplicationat~~\|[[PID implies UFD]]\|[[UFD not implies PID]]}}		\| [[Stronger than::Dedekind domain]] \|\| \|\| [[PID implies Dedekind]] \|\|[[Dedekind not implies PID]] \|\| {{intermediate notions short\|Dedekind domain\|principal ideal domain}}
	* [[Stronger than::Normal domain]]		\|-
	* [[Stronger than::Noetherian normal domain]]		\| [[Stronger than::Bezout domain]] \|\| [[integral domain]] where every [[finitely generated ideal]] is [[principal ideal\|principal]] \|\| [[PID implies Bezout]]\|\|[[Bezout not implies PID]] \|\| {{intermediate notions short\|Bezout domain\|principal ideal domain}}
	* [[~~Stronger than::One-dimensional~~ Noetherian domain]]		\|-
	* [[Stronger than::One-dimensional domain]]~~: {{proofofstrictimplicationat~~\|[[PID implies one-dimensional]]\|[[One-dimensional domain not implies PID]]}}		\| [[Stronger than::Bezout ring]] \|\| \|\| \|\| \|\| {{intermediate notions short\|Bezout ring\|principal ideal domain}}
	* [[Stronger than::Elementary divisor domain]]		\|-
	* [[Stronger than::gcd domain]]		\| [[Stronger than::Noetherian domain]] \|\| [[integral domain]] that's also a [[Noetherian ring]] \|\| [[PID implies Noetherian]] \|\| [[Noetherian not implies principal ideal]] \|\| {{intermediate notions short\|Noetherian domain\|principal ideal domain}}
	* [[Stronger than::Principal ideal ring]]		\|-
	* [[~~Stronger than::Noetherian ring~~]]		\| [[Stronger than::Noetherian ring]] \|\| \|\| \|\| \|\| {{intermediate notions short\|Noetherian ring\|principal ideal domain}}
	* [[~~Stronger than::Bezout ring~~]]		\|-
			\| [[Stronger than::Unique factorization domain]] \|\| [[integral domain]] with unique factorization \|\| [[PID implies UFD]]\|\|[[UFD not implies PID]] \|\| {{intermediate notions short\|unique factorization domain\|principal ideal domain}}
			\|-
			\| [[Stronger than::Normal domain]] \|\| [[integral domain]] that is integrally closed in its field of fractions \|\| (via UFD) \|\| (via UFD) \|\| {{intermediate notions short\|normal domain\|principal ideal domain}}
			\|-
			\| [[Stronger than::Noetherian normal domain]] \|\| [[Noetherian ring]] and [[normal domain]] \|\| \|\| \|\| {{intermediate notions short\|Noetherian normal domain\|principal ideal domain}}
			\|-
			\| [[Stronger than::One-dimensional domain]]\|\| [[integral domain]] in which every nonzero [[prime ideal]] is [[maximal ideal\|maximal]] \|\| [[PID implies one-dimensional]]\|\|[[One-dimensional domain not implies PID]] \|\| {{intermediate notions short\|one-dimensional domain\|principal ideal domain}}
			\|-
			\| [[Stronger than::One-dimensional Noetherian domain]] \|\| [[one-dimensional ring\|one-dimensional]] and [[Noetherian domain\|Noetherian]] \|\| \|\| \|\| \|\| {{intermediate notions short\|one-dimensional Noetherian domain\|principal ideal domain}}
			\|-
			\| [[Stronger than::Elementary divisor domain]] \|\| \|\| \|\| \|\| {{intermediate notions short\|elementary divisor domain\|principal ideal domain}}
			\|-
			\| [[Stronger than::gcd domain]] \|\| \|\| \|\| \|\| {{intermediate notions short\|gcd domain\|principal ideal domain}}
			\|-
			\| [[Stronger than::Principal ideal ring]] \|\| every ideal is [[principal ideal\|principal]], not necessarily a [[integral domain\|domain]] \|\| \|\| \|\| {{intermediate notions short\|principal ideal ring\|principal ideal domain}}
			\|}
	===Conjunction expressions===		===Conjunction expressions===