Integral Ideal is Ideal of Ring

Theorem
Let $J$ be a non-empty subset of the set of integers $\Z$.

Then:
 * $J$ is an integral ideal


 * $J$ is an ideal of the ring of integers $\struct {\Z, +, \times}$.
 * $J$ is an ideal of the ring of integers $\struct {\Z, +, \times}$.

Proof
Let $J \subseteq \Z$ fulfil the conditions of an integral ideal:


 * $(1): \quad n, m \in J \implies m + n \in J, m - n \in J$


 * $(2): \quad n \in J, r \in \Z \implies r n \in J$

First note that $J$ is non-empty by definition.

Then from $(1)$ we have in particular:
 * $n, m \in J \implies m - n \in J$

Thus by the One-Step Subgroup Test:


 * $\struct {J, +}$ is a subgroup of the additive group of integers $\struct {\Z, +}$.

Then from $(2)$ we have:


 * $n \in J, r \in \Z \implies r n \in J$

and by Integer Multiplication is Commutative it follows that:


 * $n \in J, r \in \Z \implies n r \in J$

Thus it is seen that the conditions are fulfilled for $J$ to be an ideal of the ring of integers $\struct {\Z, +, \times}$.

Suppose $J \subseteq \Z$ such that:


 * $(3): \quad \struct {J, +}$ is a subgroup of $\struct {\Z, +}$


 * $(4): \quad \forall n \in J: \forall r \in R: n r \in J \land r n \in J$

From $(4)$, $(2)$ immediately follows.

From $(3)$ it follows from the One-Step Subgroup Test that:


 * $\forall n, m \in J: m - n \in J$

which is a restatement of $(1)$.

Thus the conditions are fulfilled for $J$ to be an integral ideal.