\begin{definition}[Definition:Integral Ideal]
Let $J \subseteq \Z$ be a non-empty subset of the set of integers.
Let $J$ fulfil the following conditions:
:$(1): \quad n, m \in J \implies m + n \in J, m - n \in J$
:$(2): \quad n \in J, r \in \Z \implies r n \in J$
Then $J$ is an '''integral ideal'''.
\end{definition}