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Ideal
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This article is about a basic definition in commutative algebra. The article text may, however, contain more than just the basic definition. Rate its utility as a basic definition article on the talk page
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 R is a subset I that satisfies the following equivalent conditions:
- I an R-submodule of R.
- I is an Abelian group under addition and further, IR is contained inside I.
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
