I read that the global dimension of $\mathbb Z/4\mathbb Z$ is not finite. I think that it's because that $4=2\cdot 2$ and $(2,2)\neq 1$, hence $\mathbb Z/2\mathbb Z\oplus \mathbb Z/2\mathbb Z$ is not $\mathbb Z/4\mathbb Z$.
Is it the reason for this ? If it's not too complicated, I would really like to see an explanation for why the global dimension is not finite.