# 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