Ideal

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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:

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