Ideal: Difference between revisions
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==Definition | {{basicdef}} | ||
{{ring subset property}} | |||
==Definition== | |||
===Symbol-free definition=== | ===Symbol-free definition=== | ||
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For noncommutative rings, there are three notions: | For noncommutative rings, there are three notions: | ||
* [[Two-sided ideal]] | * [[nc:Two-sided ideal|Two-sided ideal]] | ||
* [[Left ideal]] | * [[nc:Left ideal|left ideal]] | ||
* [[Right ideal]] | * [[nc:Right ideal|right ideal]] | ||
==Property theory== | ==Property theory== | ||
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{{further|[[Product of ideals]]}} | {{further|[[Product of ideals]]}} | ||
Latest revision as of 16:23, 12 May 2008
This article is about a basic definition in commutative algebra. View a complete list of basic definitions in commutative algebra
This article gives a property that can be evaluated for a subset of a ring
Definition
Symbol-free definition
An ideal in a commutative unital ring (or any commutative ring) is a subset that satisfies the following equivalent conditions:
- It is a submodule of the ring viewed as a module over itself
- It is an Abelian group under addition and the product of any element in the ideal with any element in the ring lies in the ideal
- It occurs as the kernel of a ring homomorphism
Definition with symbols
An ideal in a commutative ring is a subset that satisfies the following equivalent conditions:
- an -submodule of .
- is an Abelian group under addition and further, is contained inside .
Definition for noncommutative rings
For noncommutative rings, there are three notions:
Property theory
Intersection
An arbitrary intersection of ideals is again an ideal.
Sum
The Abelian group generated by any family of ideals (when treated as Abelian groups) is itself an ideal, and is in fact the smallest ideal generated by them.
Product
Further information: Product of ideals