Definition:Ideal of Ring

Definition
Let $\struct {R, +, \circ}$ be a ring.

Let $\struct {J, +}$ be a subgroup of $\struct {R, +}$.

Then $J$ is an ideal of $R$ :
 * $\forall j \in J: \forall r \in R: j \circ r \in J \land r \circ j \in J$

that is, :
 * $\forall r \in R: J \circ r \subseteq J \land r \circ J \subseteq J$

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.

Some sources refer to such a two-sided ideal as a normal subring, in apposition with the concept of a normal subgroup.

Also see

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