Congruence reflection for Frobenius substitution (${\varphi }^{-1}(p^{k}R)=p^{k}R$)

For $R={\mathbb{Z}}_p[[T]]$ and $\varphi (f)=f((1+T{)}^p-1)$, reduction modulo $p$ is the injective substitution $\overline{f}(T)\mapsto \overline{f}(T^p)$. Thus $\varphi (f)\in pR$ implies $f\in pR$. Dividing by $p$ and repeating proves the displayed equality for every $k\ge 1$. In particular, a ring congruence of Frobenius substitutions on cyclotomic power series reflects back to the original formal power series.