Ideal with prime radical: Difference between revisions
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==Definition== | ==Definition== | ||
An [[ideal]] in a [[commutative unital ring]] is said to have '''prime radical''' if its [[defining ingredient::radical of an ideal|radical]] is a [[defining ingredient::prime ideal]]. | |||
{{curing-ideal property}} | |||
==Relation with other properties== | ==Relation with other properties== | ||
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===Stronger properties=== | ===Stronger properties=== | ||
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! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |||
|- | |||
| [[Weaker than::prime ideal]] || product of two elements in the ideal implies at least one of them is in the ideal || follows from the fact that [[prime implies radical]], so a prime ideal equals its own radical || [[prime radical not implies prime]] || {{intermediate notions short|ideal with prime radical|prime ideal}} | |||
|- | |||
| [[Weaker than::primary ideal]] || product of two elements in the ideal implies either the first or a power of the second is in the ideal || [[primary implies prime radical]] || [[prime radical not implies primary]] || {{intermediate notions short|ideal with prime radical|prime ideal}} | |||
|- | |||
| [[Weaker than::power of a prime ideal]] || product of a [[prime ideal]] with itself finitely many times || || || | |||
|- | |||
| [[Weaker than::ideal with maximal radical]] || [[radical of an ideal|radical]] is a [[maximal ideal]] || || || | |||
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Latest revision as of 17:48, 18 December 2011
Definition
An ideal in a commutative unital ring is said to have prime radical if its radical is a prime ideal.
This article defines a property of an ideal in a commutative unital ring |View other properties of ideals in commutative unital rings
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| prime ideal | product of two elements in the ideal implies at least one of them is in the ideal | follows from the fact that prime implies radical, so a prime ideal equals its own radical | prime radical not implies prime | click here |
| primary ideal | product of two elements in the ideal implies either the first or a power of the second is in the ideal | primary implies prime radical | prime radical not implies primary | click here |
| power of a prime ideal | product of a prime ideal with itself finitely many times | |||
| ideal with maximal radical | radical is a maximal ideal |