# Subgroup of Integers is Ideal

## Theorem

Let $\struct {\Z, +}$ be the additive group of integers.

Every subgroup of $\struct {\Z, +}$ is an ideal of the ring $\struct {\Z, +, \times}$.

### Corollary

Every subring of $\struct {\Z, +, \times}$ is an ideal of the ring $\struct {\Z, +, \times}$.

## Proof

Let $H$ be a subgroup of $\struct {\Z, +}$.

Let $n \in \Z, h \in H$.

Then from the definition of cyclic group and Negative Index Law for Monoids:

$n h = n \cdot h \in \gen h \subseteq H$

The result follows.

$\blacksquare$