Subgroup of Integers is Ideal

Theorem
Let $$\left({\Z, +}\right)$$ be the Additive Group of Integers.

Every subgroup of $$\left({\Z, +}\right)$$ is an ideal of the ring $$\left({\Z, +, \times}\right)$$.

Corollary
Every subring of $$\left({\Z, +, \times}\right)$$ is an ideal of the ring $$\left({\Z, +, \times}\right)$$.

Proof
Let $$H$$ be a subgroup of $$\left({\Z, +}\right)$$.

Let $$n \in \Z, h \in H$$. Then from the definition of cyclic group and Negative Index Law for Monoids:


 * $$n h = n \cdot h \in \left \langle {h} \right \rangle \subseteq H$$

The result follows.

Proof of Corollary
Follows directly from Subrings of the Integers and the above result.