I am trying to classify, $\frac{\mathbb{Z} \times \mathbb{Z}}{\langle(n,n)\rangle}$, $n=1,2,3,\cdots, $ using fundamental theorem of finitely generated abelian groups.

Inspire by @user134824 in [Classifying the factor group $(\mathbb{Z} \times \mathbb{Z})/\langle (2, 2) \rangle$](https://math.stackexchange.com/questions/720900/classifying-the-factor-group-mathbbz-times-mathbbz-langle-2-2-ran), I construct $\varphi: \mathbb{Z} \times \mathbb{Z} \rightarrow \mathbb{Z} \times \mathbb{Z}\_n$ by $(k,l) \mapsto (k-l,l)$. Then its kernel is $\langle(n,n)\rangle$ and from the 1st isomorphism theorem, $\mathbb{Z} \times \mathbb{Z} \simeq \mathbb{Z}\_n \times \mathbb{Z}$.

But how I can understand this from the fundamental theorem of finitely generated abelian groups?