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 $m$.

Proof
Consider the map $\phi : \struct {m \Z, +} \times \struct {n \Z, +} \mapsto \struct {\Z, +} \times \struct {\Z, +}$ defined by:
 * $\phi (m c, n d) = (c, d) \quad \forall c, d \in \Z$

which is an isomorphism.

Hence, $\struct {m \Z, +} \times \struct {n \Z, +}$ is a free abelian group of rank $2$.

Therefore, any singly generated subgroup, for example, $\set{(x, 0): x \in \Z}$ is a subgroup not in the form $\struct {m \Z, +} \times \struct {n \Z, +}$.