Definition:Ideal of Ring

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

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

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

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

Proper Ideal
A proper ideal $$J$$ of $$\left({R, +, \circ}\right)$$ is an ideal of $$R$$ such that $$J$$ is a proper subset of $$R$$.

That is, such that $$J \subset R$$ and $$J \ne R$$.