Definition:Ideal of Ring

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

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

$$\forall s \in S: \forall r \in R: s \circ r \in S \land r \circ s \in S$$

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

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