Radical ideal - Revision history

Vipul: /* Important ring types */

2011-12-18T17:50:34Z

Important ring types

← Older revision		Revision as of 17:50, 18 December 2011
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	! Property of commutative unital rings !! Characterization of radical ideals in such a ring !! Examples		! Property of commutative unital rings !! Characterization of radical ideals in such a ring !! Examples
	\|-		\|-
	\| [[principal ideal domain]] (i.e., an [[integral domain]] in which every ideal is a [[principal ideal]]) \|\| either the zero ideal or a [[principal ideal]] generated by a square-free element, i.e., the generating element should be a product of distinct primes \|\| In the [[ring of integers]], the radical ideals are precisely the zero ideal and the ideals generated by square-free numbers. Thus, for instance, <math>6\mathbb{Z}</math> and <math>30\mathbb{Z}</math> are radical ideals in <math>\mathbb{Z}</math> but <math>24\mathbb{Z}</math> is not.<br>In the (univariate) [[polynomial ring over a field]], the radical ideals are precisely the zero ideal and the ideals generated by polynomials with no repeated roots over the algebraic closure, i.e., polynomials that are relatively prime to their derivative polynomial. Thus, for instance, <math>\langle x^2 + 1 \rangle</math> and <math>x^2 - 1</math> are radical ideals in <math>\mathbb{R}[x]</math> but <math>\langle x^2 \rangle</math> is not.		\| [[principal ideal domain]] (i.e., an [[integral domain]] in which every ideal is a [[principal ideal]]) \|\| either the zero ideal or a [[principal ideal]] generated by a square-free element, i.e., the generating element should be a product of distinct primes \|\| In the [[ring of integers]], the radical ideals are precisely the zero ideal and the ideals generated by square-free numbers. Thus, for instance, <math>6\mathbb{Z}</math> and <math>30\mathbb{Z}</math> are radical ideals in <math>\mathbb{Z}</math> but <math>24\mathbb{Z}</math> is not.<br>In the (univariate) [[polynomial ring over a field]], the radical ideals are precisely the zero ideal and the ideals generated by square-free polynomials, which can be described as polynomials with no repeated roots over the algebraic closure, i.e., polynomials that are relatively prime to their derivative polynomial. Thus, for instance, <math>\langle x^2 + 1 \rangle</math> and <math>x^2 - 1</math> are radical ideals in <math>\mathbb{R}[x]</math> but <math>\langle x^2 \rangle</math> is not.
	\|-		\|-
	\| [[Dedekind domain]] (roughly speaking, admits unique factorization of ideals into prime ideals) \|\| either zero or a product of ''distinct'' prime ideals \|\| {{fillin}}		\| [[Dedekind domain]] (roughly speaking, admits unique factorization of ideals into prime ideals) \|\| either zero or a product of ''distinct'' prime ideals \|\| {{fillin}}

Vipul: /* Important ring types */

2011-12-18T17:42:00Z

Important ring types

← Older revision		Revision as of 17:42, 18 December 2011
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	! Property of commutative unital rings !! Characterization of radical ideals in such a ring !! Examples		! Property of commutative unital rings !! Characterization of radical ideals in such a ring !! Examples
	\|-		\|-
	\| [[principal ideal domain]] (i.e., an [[integral domain]] in which every ideal is a [[principal ideal]]) \|\| either the zero ideal or a [[principal ideal]] generated by a square-free element, i.e., the generating element should be a product of distinct primes \|\| In the [[ring of integers]], the radical ideals are precisely the zero ideal and the ideals generated by square-free numbers. Thus, for instance, <math>6\mathbb{Z}</math> and <math>30\mathbb{Z}</math> are radical ideals in <math>\mathbb{Z}</math> but <math>24\mathbb{Z}</math> is not.<br>In the (univariate) [[polynomial ring over a field]], the radical ideals are precisely the zero ideal and the ideals generated by polynomials with no repeated roots over the algebraic closure, i.e., polynomials that are relatively prime to their derivative polynomial.		\| [[principal ideal domain]] (i.e., an [[integral domain]] in which every ideal is a [[principal ideal]]) \|\| either the zero ideal or a [[principal ideal]] generated by a square-free element, i.e., the generating element should be a product of distinct primes \|\| In the [[ring of integers]], the radical ideals are precisely the zero ideal and the ideals generated by square-free numbers. Thus, for instance, <math>6\mathbb{Z}</math> and <math>30\mathbb{Z}</math> are radical ideals in <math>\mathbb{Z}</math> but <math>24\mathbb{Z}</math> is not.<br>In the (univariate) [[polynomial ring over a field]], the radical ideals are precisely the zero ideal and the ideals generated by polynomials with no repeated roots over the algebraic closure, i.e., polynomials that are relatively prime to their derivative polynomial. Thus, for instance, <math>\langle x^2 + 1 \rangle</math> and <math>x^2 - 1</math> are radical ideals in <math>\mathbb{R}[x]</math> but <math>\langle x^2 \rangle</math> is not.
	\|-		\|-
	\| [[Dedekind domain]] (roughly speaking, admits unique factorization of ideals into prime ideals) \|\| either zero or a product of ''distinct'' prime ideals \|\| {{fillin}}		\| [[Dedekind domain]] (roughly speaking, admits unique factorization of ideals into prime ideals) \|\| either zero or a product of ''distinct'' prime ideals \|\| {{fillin}}

Vipul: /* Important ring types */

2011-12-18T17:41:07Z

Important ring types

← Older revision		Revision as of 17:41, 18 December 2011
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	! Property of commutative unital rings !! Characterization of radical ideals in such a ring !! Examples		! Property of commutative unital rings !! Characterization of radical ideals in such a ring !! Examples
	\|-		\|-
	\| [[principal ideal domain]] (i.e., an [[integral domain]] in which every ideal is a [[principal ideal]]) \|\| either the zero ideal or a [[principal ideal]] generated by a square-free element, i.e., the generating element should be a product of distinct primes \|\| In the [[ring of integers]], the radical ideals are precisely the zero ideal and the ideals generated by square-free numbers.<br>In the (univariate) [[polynomial ring over a field]], the radical ideals are precisely the zero ideal and the ideals generated by polynomials with no repeated roots over the algebraic closure, i.e., polynomials that are relatively prime to their derivative polynomial.		\| [[principal ideal domain]] (i.e., an [[integral domain]] in which every ideal is a [[principal ideal]]) \|\| either the zero ideal or a [[principal ideal]] generated by a square-free element, i.e., the generating element should be a product of distinct primes \|\| In the [[ring of integers]], the radical ideals are precisely the zero ideal and the ideals generated by square-free numbers. Thus, for instance, <math>6\mathbb{Z}</math> and <math>30\mathbb{Z}</math> are radical ideals in <math>\mathbb{Z}</math> but <math>24\mathbb{Z}</math> is not.<br>In the (univariate) [[polynomial ring over a field]], the radical ideals are precisely the zero ideal and the ideals generated by polynomials with no repeated roots over the algebraic closure, i.e., polynomials that are relatively prime to their derivative polynomial.
	\|-		\|-
	\| [[Dedekind domain]] (roughly speaking, admits unique factorization of ideals into prime ideals) \|\| either zero or a product of ''distinct'' prime ideals \|\| {{fillin}}		\| [[Dedekind domain]] (roughly speaking, admits unique factorization of ideals into prime ideals) \|\| either zero or a product of ''distinct'' prime ideals \|\| {{fillin}}

Vipul: /* Relation with other properties */

2011-12-18T17:39:53Z

Relation with other properties

← Older revision		Revision as of 17:39, 18 December 2011
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	===Stronger properties===		===Stronger properties===

	* [[Weaker than::Maximal ideal]]		{\| class="sortable" border="1"
	* [[Weaker than::Prime ideal]]		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions !! Collapse
			\|-
			\| [[Weaker than::Maximal ideal]] \|\| proper ideal not contained in any bigger proper ideal<br>quotient is a [[field]] \|\| (via prime ideal) \|\| (via prime ideal) \|\| \|\| ?
			\|-
			\| [[Weaker than::Prime ideal]] \|\| quotient is an [[integral domain]] \|\| [[prime implies radical]] \|\| [[radical not implies prime]] \|\| \|\| ?
			\|-
			\| [[Weaker than::Intersection of maximal ideals]] \|\| intersection of [[maximal ideal]]s \|\| \|\| \|\| \|\| [[Jacobson ring]]
			\|}

	===Incomparable properties===		===Incomparable properties===

Vipul at 17:36, 18 December 2011

2011-12-18T17:36:22Z

@@ Line 20: / Line 20: @@
 {{curing-ideal property}}
 {{quotient is a|reduced ring}}
 ==Relation with other properties==

Vipul: /* Metaproperties */

2011-12-18T17:29:45Z

Metaproperties

← Older revision		Revision as of 17:29, 18 December 2011
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	==Metaproperties==		==Metaproperties==

	{{intersection-closed ~~ideal~~ property}}		{\| class="sortable" border="1"
			! Metaproperty !! Satisfied? !! Proof !! Statement with symbols
	~~An arbitrary~~ intersection of radical ideals is ~~again~~ a radical ideal.		\|-
			\| [[satisfies metaproperty::intersection-closed property of ideals in commutative unital rings]] \|\| Yes \|\| [[intersection of radical ideals is radical]] \|\| Suppose <math>I_s, s \in S</math> is a (possibly finite, possibly infinite) collection of radical ideals in a commutative unital ring <math>R</math>. Then the intersection <math>\bigcap_{s \in S} I_s</math> is also a radical ideal in <math>R</math>.
	{{contraction-closed ~~ideal~~ property}}		\|-
			\| [[satisfies metaproperty::contraction-closed property of ideals in commutative unital rings]] \|\| Yes \|\| {{fillin}} \|\| If <math>f:R \to S</math> is a [[homomorphism of commutative unital rings]], and <math>I</math> is a radical ideal in <math>S</math>, then <math>f^{-1}(I)</math> is a radical ideal in <math>R</math>.
	If <math>f:R \to S</math> is a [[homomorphism of commutative unital rings]], and <math>I</math> is a radical ideal in <math>S</math>, then <math>f^{-1}(I)</math> is a radical ideal in <math>R</math>.		\|-
			\| [[satisfies metaproperty::intermediate subring condition for ideals]] \|\| Yes \|\| {{fillin}} \|\| Suppose <math>I</math> is a radical ideal in a commutative unital ring <math>R</math> and <math>S</math> is a unital subring of <math>R</math> that contains <math>I</math>. Then, <math>I</math> is also a radical ideal in <math>S</math>.
	{{~~intringcondn ideal~~}}		\|-
			\| [[satisfies metaproperty::transfer condition for ideals]] \|\| Yes \|\| {{fillin}} \|\| If <math>I</math> is a radical ideal in <math>R</math>, and <math>S</math> is a subring of <math>R</math>, then <math>I \cap S</math> is a radical ideal in <math>S</math>.
	A radical ideal of a ring is also a radical ideal in ~~any intermediate subring. This corresponds to the fact that any subring of a reduced ring is again reduced~~.		\|}

	{{~~transfercondn ideal~~}}

	If <math>I</math> is a radical ideal in <math>R</math>, and <math>S</math> is a subring of <math>R</math>, then <math>I \cap S</math> is a radical ideal in <math>S</math>.

Vipul: /* Equivalent definitions in tabular format */

2011-12-18T17:24:54Z

Equivalent definitions in tabular format

← Older revision		Revision as of 17:24, 18 December 2011
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	\| 1 \|\| closed under taking roots \|\| whenever a power of an element in the ring lies inside that ideal, the element itself lies inside that ideal \|\| For any <math>a \in R</math> and any positive integer <math>n</math>, if <math>a^n \in I</math>, then <math>a \in I</math>.		\| 1 \|\| closed under taking roots \|\| whenever a power of an element in the ring lies inside that ideal, the element itself lies inside that ideal \|\| For any <math>a \in R</math> and any positive integer <math>n</math>, if <math>a^n \in I</math>, then <math>a \in I</math>.
	\|-		\|-
	\| 2 \|\| ~~quotient ring is reduced~~ \|\| ~~the [[quotient ring]] by the ideal has trivial~~ [[defining ingredient::~~nilradical~~]] ~~(that is, it is a [[defining ingredient::reduced~~ ring~~]])~~ \|\| ~~the quotient ring~~ <math>R/I~~</math> is a reduced ring: whenever <math>x~~ \~~in R/~~I</math> ~~and~~ <math>n</math> ~~is a~~ positive ~~integer such that <math>x^n = 0</math>, then <math>x = 0</math>~~.		\| 2 \|\| equals its own radical \|\| it equals its own [[defining ingredient::radical of an ideal\|radical]] in the whole ring \|\| <math>I = \sqrt{I}</math> where <math>\sqrt{}</math> denotes the [[radical of an ideal]], i.e., the set of all elements for which some positive power lies inside the ideal.
	\|-		\|-
	\| 3 \|\| intersection of prime ideals \|\| it can be expressed as an intersection of [[defining ingredient::prime ideal]]s. The intersection is allowed to be finite or infinite. An empty intersection, which would give the whole ring, is also allowed. \|\| There exists a collection <math>P_s, s \in S</math> of prime ideals indexed by a set <math>S</math> such that <math>I = \bigcap_{s \in S} P_s</math>. <math>S</math> is allowed to be finite or infinite, and is also allowed to be empty.		\| 3 \|\| quotient ring is reduced \|\| the [[quotient ring]] by the ideal has trivial [[defining ingredient::nilradical]] (that is, it is a [[defining ingredient::reduced ring]]) \|\| the quotient ring <math>R/I</math> is a reduced ring: whenever <math>x \in R/I</math> and <math>n</math> is a positive integer such that <math>x^n = 0</math>, then <math>x = 0</math>.
			\|-
			\| 4 \|\| intersection of prime ideals \|\| it can be expressed as an intersection of [[defining ingredient::prime ideal]]s. The intersection is allowed to be finite or infinite. An empty intersection, which would give the whole ring, is also allowed. \|\| There exists a collection <math>P_s, s \in S</math> of prime ideals indexed by a set <math>S</math> such that <math>I = \bigcap_{s \in S} P_s</math>. <math>S</math> is allowed to be finite or infinite, and is also allowed to be empty.
	\|}		\|}

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	{{curing-ideal property}}		{{curing-ideal property}}
	{{quotient is a\|reduced ring}}		{{quotient is a\|reduced ring}}

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

Vipul at 17:15, 18 December 2011

2011-12-18T17:15:56Z

← Older revision		Revision as of 17:15, 18 December 2011
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	~~{{basicdef}}~~
	~~{{curing-ideal property}}~~
	~~{{quotient is a\|reduced ring}}~~

	==Definition==		==Definition==

	~~An [[ideal]]~~ in ~~a [[commutative unital ring]] is termed a '''radical ideal''' if it satisfies the following equivalent conditions:~~		===Equivalent definitions in tabular format===

	* Whenever a power of an element in the ring lies inside that ideal, the element itself lies inside that ideal		{\| class="sortable" border="1"
	* The [[quotient ring]] by the ideal has trivial [[nilradical]] (that is, it is a [[reduced ring]])		! No. !! Shorthand !! An [[ideal]] in a [[commutative unital ring]] is termed a radical ideal if ... !! An [[ideal]] <math>I</math> in a [[commutative unital ring]] <math>R</math> is termed a radical ideal if ...
	* It is an intersection of [[prime ideal]]s		\|-
			\| 1 \|\| closed under taking roots \|\| whenever a power of an element in the ring lies inside that ideal, the element itself lies inside that ideal \|\| For any <math>a \in R</math> and any positive integer <math>n</math>, if <math>a^n \in I</math>, then <math>a \in I</math>.
			\|-
			\| 2 \|\| quotient ring is reduced \|\| the [[quotient ring]] by the ideal has trivial [[defining ingredient::nilradical]] (that is, it is a [[defining ingredient::reduced ring]]) \|\| the quotient ring <math>R/I</math> is a reduced ring: whenever <math>x \in R/I</math> and <math>n</math> is a positive integer such that <math>x^n = 0</math>, then <math>x = 0</math>.
			\|-
			\| 3 \|\| intersection of prime ideals \|\| it can be expressed as an intersection of [[defining ingredient::prime ideal]]s. The intersection is allowed to be finite or infinite. An empty intersection, which would give the whole ring, is also allowed. \|\| There exists a collection <math>P_s, s \in S</math> of prime ideals indexed by a set <math>S</math> such that <math>I = \bigcap_{s \in S} P_s</math>. <math>S</math> is allowed to be finite or infinite, and is also allowed to be empty.
			\|}

	Note that unlike for the definition of prime ideals, we do ''not'' require a radical ideal to be proper, so the whole ring is always a radical ideal in itself.		Note that unlike for the definition of prime ideals, we do ''not'' require a radical ideal to be proper, so the whole ring is always a radical ideal in itself.

			{{basicdef}}
			{{curing-ideal property}}
			{{quotient is a\|reduced ring}}
	==Relation with other properties==		==Relation with other properties==

Vipul: /* Incomparable properties */

2009-01-17T20:22:05Z

Incomparable properties

← Older revision		Revision as of 20:22, 17 January 2009
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	===Incomparable properties===		===Incomparable properties===

	* [[~~Stronger than::~~Irreducible ideal]]		* [[Irreducible ideal]]
	* [[~~Stronger than::~~Primary ideal]]		* [[Primary ideal]]

	==Metaproperties==		==Metaproperties==

Vipul: /* Relation with other properties */

2009-01-17T20:21:46Z

Relation with other properties

← Older revision		Revision as of 20:21, 17 January 2009
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	===Stronger properties===		===Stronger properties===

	* [[Maximal ideal]]		* [[Weaker than::Maximal ideal]]
	* [[Prime ideal]]		* [[Weaker than::Prime ideal]]

	===Incomparable properties===		===Incomparable properties===

	* [[Irreducible ideal]]		* [[Stronger than::Irreducible ideal]]
	* [[Primary ideal]]		* [[Stronger than::Primary ideal]]

	==Metaproperties==		==Metaproperties==