Definition:Ideal of Ring/Proper Ideal

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

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 $J \subseteq R$ and $J \ne R$.