We know that for all $n \neq 0$, $\mathbb{Z}/n\mathbb{Z}$ is an injective module over itself. But $\mathbb{Z}/n\mathbb{Z}$ is also a PID, then since it is injective by Baer's criterion $\overline{x}(\mathbb{Z}/n\mathbb{Z})= \mathbb{Z}/n\mathbb{Z}$ for all non zero $\overline{x} \in \mathbb{Z}/n\mathbb{Z}$.
If we consider the example $\mathbb{Z}/4\mathbb{Z}$ and $\overline{x}=\overline{2}$ then $\overline{2}(\mathbb{Z}/4\mathbb{Z})=\{\overline{0},\overline{2}\}$ which is properly contained in $\mathbb{Z}/4\mathbb{Z}$. What is the wrong thing I am doing? Please help me.
The crucial error is that $\mathbb{Z}/n\mathbb{Z}$ is not a PID for all $n$. Indeed, it's not even an integral domain unless $n$ is prime.