Ideal

Definition for commutative rings

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:

• Two-sided ideal
• left ideal
• right ideal

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