# Ideal

From Commalg

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`