Injective module but not flat module?

Here's an example from abelian groups $=$ $\mathbb{Z}$-modules: $\mathbb{Q}/\mathbb{Z}$ is an injective $\mathbb{Z}$-module because it's a divisible abelian group, but it's not a flat $\mathbb{Z}$-module because $\require{cancel}\xcancel{\mathbb{Q}/\mathbb{Z}\otimes_{\mathbb{Z}}\mathbb{Z}=0}$ (... sorry for the mistake ...) $\mathbb{Q}/\mathbb{Z}\otimes_{\mathbb{Z}}\mathbb{Q}=0$.