\begin{definition}[Definition:Ideal of Ring]
Let $\struct {R, +, \circ}$ be a ring.
Let $\struct {J, +}$ be a subgroup of $\struct {R, +}$.
Then $J$ is an '''ideal of $R$''' {{iff}}:
:$\forall j \in J: \forall r \in R: j \circ r \in J \land r \circ j \in J$
that is, {{iff}}:
:$\forall r \in R: J \circ r \subseteq J \land r \circ J \subseteq J$
The letter $J$ is frequently used to denote an '''ideal'''.
\end{definition}