Radical ideal: Difference between revisions
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{{curing-ideal property}} | |||
An [[ideal]] in a [[commutative unital ring]] | {{quotient is a|reduced ring}} | ||
==Definition== | |||
An [[ideal]] in a [[commutative unital ring]] is termed a '''radical ideal''' if it satisfies the following equivalent conditions: | |||
* Whenever a power of an element in the ring lies inside that ideal, the element itself lies inside that ideal | * Whenever a power of an element in the ring lies inside that ideal, the element itself lies inside that ideal | ||
* The [[quotient ring]] by the ideal has trivial [[nilradical]] (that is, it is a [[reduced ring]]) | * The [[quotient ring]] by the ideal has trivial [[nilradical]] (that is, it is a [[reduced ring]]) | ||
== | ==For non-commutative rings== | ||
There are the following definitions: | There are the following definitions: | ||
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* [[Radical ideal (noncommutative rings)]] | * [[Radical ideal (noncommutative rings)]] | ||
* [[Semiprime ideal]] | * [[Semiprime ideal]] | ||
Revision as of 09:32, 7 August 2007
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
Definition
An ideal in a commutative unital ring is termed a radical ideal if it satisfies the following equivalent conditions:
- Whenever a power of an element in the ring lies inside that ideal, the element itself lies inside that ideal
- The quotient ring by the ideal has trivial nilradical (that is, it is a reduced ring)
For non-commutative rings
There are the following definitions: