Amalgamated product $\mathbb{Z}*_{\mathbb{Z}} 0$

Solution 1:

Why do you think that taking an amalgamated product with $0$ gives $0$? What that operation does is identify the image of multiplication by $\mathbb Z\longrightarrow \mathbb Z$ in $\mathbb Z$ with $0$, so it is indeed the quotient you describe.

Concretely, if you have groups $G_1$ and $G_2$ and maps $f_2 : G_2\longleftarrow H\longrightarrow G_1 : f_1$, then the amalgamated product $G_1\ast_H G_2$ is the quotient of the free product $G_1\ast G_2$ by the relation that identifies the elements in $G_2$ coming from $f_2$ with the elements in $G_1$ coming from $f_1$.

In your case, we have that $\mathbb Z\ast 0 = \mathbb Z$ and that you are identifying $3\mathbb Z$ with $0$.