Nilradical: Difference between revisions
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{{ideal-defining function}} | {{curing-ideal-defining function}} | ||
==Definition== | ==Definition== | ||
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* It is the radical of zero. | * It is the radical of zero. | ||
* It is the set of [[nilpotent element]]s | * It is the set of [[nilpotent element]]s | ||
===Equivalence of definitions=== | |||
For a proof of the equivalence of definitions, see [[nilradical is smallest radical ideal]] and [[nilradical equals intersection of all prime ideals]] (the remaining equivalences are direct from definitions). | |||
Latest revision as of 16:27, 12 May 2008
This article defines an ideal-defining function, viz a rule that inputs a commutative unital ring and outputs an ideal of that ring
Definition
Symbol-free definition
The nilradical of a commutative unital ring is defined as the subset that satisfies the following equivalent conditions:
- It is the intersection of all prime ideals
- It is the intersection of all radical ideals
- It is the radical of zero.
- It is the set of nilpotent elements
Equivalence of definitions
For a proof of the equivalence of definitions, see nilradical is smallest radical ideal and nilradical equals intersection of all prime ideals (the remaining equivalences are direct from definitions).