$\mathbb{Z}\oplus\mathbb{Z}=\{(i,j):i,j\in\mathbb{Z}\}$ and we can think two equivalence classes namely;$$c\_1=\{(i,0):i\in\mathbb{Z}\}$$ and $$c\_2=\{(0,i):i\in\mathbb{Z}\}$$ such that $$c-c'\in\mathbb{Z}\ \forall c,c'\in c\_{1}$$ and $$c-c'\in\mathbb{Z}\ \forall c,c'\in c\_{2}$$ thus
$$(\mathbb{Z}\oplus\mathbb{Z})/\mathbb{Z}=\{[c\_1],[c\_2]\}\cong\mathbb{Z}\_2$$ Is this correct?