Wrong example of not abelian homotopy category? I was reading the book **Triangulated Categories** by Thorsten Holm, Peter Jørgensen, Raphaël Rouquier. I found in the book the example below. $ is not abelian and in order to do this he takes a zero-arrow $f$ and he claims that $f$ has not kernel. Is this example wrong? If I take the zero arrow $0\colon A\longrightarrow B$ in any additive category, then is it true that it has kernel given by the identity map of $A$?

Yes, this example is just wrong. You are correct that the identity map on $A$ is always a kernel of the zero map $A\to B$. Their argument seems to assume that $r\mathbb{Z}$ cannot be all of $\mathbb{Z}$, but this is of course incorrect if $r=\pm 1$, which is exactly what happens for the identity map.