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.