Definition:Ideal of Ring

Definition
Let $\left({R, +, \circ}\right)$ be a ring.

Let $\left({J, +}\right)$ be a subgroup of $\left({R, +}\right)$.

Then $J$ is an ideal of $R$ :


 * $\forall j \in J: \forall r \in R: j \circ r \in J \land r \circ j \in J$

The letter $J$ is frequently used to denote an ideal.

Right Ideal
It follows that in a commutative ring, a left ideal, a right ideal and an ideal are the same thing.

Also known as
An ideal can also be referred to as a two-sided ideal to distinguish it from a left ideal and a right ideal.

Some sources use $I$ to denote an ideal, but on this can be too easily conflated with an identity mapping.

Also see

 * Definition:Prime Ideal of Ring
 * Definition:Maximal Ideal