An example of a module that is not injective I know that since $\mathbb Z$ is a PID every free module is projective and conversely.
Hence since $\mathbb Q$ is not free as a $\mathbb Z$-module then it is not projective.

But is $\mathbb Q$ an injective $\mathbb Z$ module? Does there exist some similar result like above to prove this fact?

If not then what is an example of a module that is not injective?

The injective $\mathbb{Z}$-modules are precisely the divisible abelian groups (in particular, $\mathbb{Q}$ is an injective $\mathbb{Z}$-module). Now, the abelian groups $\mathbb{Z}$ and $\mathbb{Z}/n\mathbb{Z}$ are not divisible, and hence give us examples of non-injective $\mathbb{Z}$-modules.
