Subgroups of Cartesian Product of Additive Group of Integers

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

Let $m, n \in \Z_{> 0}$ be (strictly) positive integers.

Let $\struct {\Z \times \Z, +}$ denote the Cartesian product of $\struct {\Z, +}$ with itself.

The subgroups of $\struct {\Z \times \Z, +}$ are not all of the form:
 * $\struct {m \Z, +} \times \struct {n \Z, +}$

where $\struct {m \Z, +}$ denotes the additive group of integer multiples of $m$.