I am trying to prove that $\mathbb{Z}/p\mathbb{Z}$ is not a flat $\mathbb{Z}/p\mathbb{Z}$-module. The reasoning I have is the following.
We have an exact sequence $0 \to \mathbb{Z}/p\mathbb{Z} \to\mathbb{Z}/p^2\mathbb{Z} \to \mathbb{Z}/p\mathbb{Z} \to 0$ where the first map is multiplication by $p$ and the last map is the cokernel of the first map. This sequence does not split and hence $\mathbb{Z}/p\mathbb{Z}$ is not a projective $\mathbb{Z}/p^2\mathbb{Z}$-module. However every finitely presented flat module is projective. Clearly $\mathbb{Z}/p\mathbb{Z}$ is finitely presented and not projective. Hence it is not flat.
1.) Is my reasoning correct ?
2.) Can we write down an exact sequence of $\mathbb{Z}/p^2\mathbb{Z}$-modules which does not remain exact after tensoring with $\mathbb{Z}/p\mathbb{Z}$ ?
Thanks !