Let $K$ be a field and $\phi: K[t] \to K[t], t \mapsto t^n$, $n \ge 2$. This $\phi$ makes the second $K[t]$ to a $K[t]$-module by $a,b \in K[t]: a \cdot b:=\phi(a)b$.
Why this $K[t]$-module structure on $K[t]$ is flat?
If yes, why? And how far this example can be generalized under preservation of flatness? That is, for example is the $R[t]$-module $R[t]$ with same $\phi$ as above flat, it we now consider an arbitrary ring $R$ instead a field $K$ as above.
Other generalization is if we consider a polynomial map $\psi: K[t_1,...t_n] \to K[z_1,...,z_m]$. Is there any criterion when $\psi$ induces a flat $K[t_1,...t_n]$-module structure on $K[z_1,...,z_m]$?