Radical ideal: Difference between revisions

Revision as of 17:24, 18 December 2011

Definition

Equivalent definitions in tabular format

No.	Shorthand	An ideal in a commutative unital ring is termed a radical ideal if ...	An ideal $I$ in a commutative unital ring $R$ is termed a radical ideal if ...
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 $a\in R$ and any positive integer $n$ , if $a^{n}\in I$ , then $a\in I$ .
2	equals its own radical	it equals its own radical in the whole ring	$I={\sqrt {I}}$ where ${\sqrt {}}$ denotes the radical of an ideal, i.e., the set of all elements for which some positive power lies inside the ideal.
3	quotient ring is reduced	the quotient ring by the ideal has trivial nilradical (that is, it is a reduced ring)	the quotient ring $R/I$ is a reduced ring: whenever $x\in R/I$ and $n$ is a positive integer such that $x^{n}=0$ , then $x=0$ .
4	intersection of prime ideals	it can be expressed as an intersection of prime ideals. 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 $P_{s},s\in S$ of prime ideals indexed by a set $S$ such that $I=\bigcap _{s\in S}P_{s}$ . $S$ 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.

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 an ideal in a ring is equivalent to the property of the quotient ring being a/an: reduced ring | View other quotient-determined properties of ideals in commutative unital rings

Relation with other properties

Stronger properties

Incomparable properties

Metaproperties

Intersection-closedness

This property of ideals in commutative unital rings is intersection-closed: an arbitrary intersection of ideals with this property, also has this property

An arbitrary intersection of radical ideals is again a radical ideal.

Contraction-closedness

This property of ideals in commutative unital rings is contraction-closed: a contraction of an ideal with this property, also has this property
View other contraction-closed properties of ideals in commutative unital rings

If $f:R\to S$ is a homomorphism of commutative unital rings, and $I$ is a radical ideal in $S$ , then $f^{-1}(I)$ is a radical ideal in $R$ .

Intermediate subring condition

This property of ideals satisfies the intermediate subring condition for ideals: if an ideal has this property in the whole ring, it also has this property in any intermediate subring
View other properties of ideals satisfying the intermediate subring condition

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.

Transfer condition

This property of ideals satisfies the transfer condition for ideals: if an ideal satisfies the property in the ring, its intersection with any subring satisfies the property inside that subring

If $I$ is a radical ideal in $R$ , and $S$ is a subring of $R$ , then $I\cap S$ is a radical ideal in $S$ .

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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>.
 |-
-| 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}}
 {{quotient is a|reduced ring}}
 ==Relation with other properties==