In Silverman's Arithmetic of Elliptic Curves, Corollary IV.4.4 states that (for an arbitrary ring $R$),
Let $\mathcal{F}/R$ be a formal group and let $p\in \mathbb{Z}$ be a prime. There are power series $f(T), g(T)\in R[\![ T ]\!]$ with $f(0) = g(0) = 0$ such that $[p](T) = pf(T) + g(T^p).$
In the proof, Silverman needs that if the formal derivative $[p]'(T)$ is in $pR[\![T]\!]$, then every term $aT^n$ in the series $[p](T)$ is such that either $a\in pR$ or $p\mid n$.
Question 1. But I think that this argument is valid if and only if $p$ is a prime in $R$. Could anyone tell me why Silverman's proof is valid in general? (I did not find this in his errata of AEC.) Thanks in advance!
Below is a screenshot of the proof: 
Question 2. (if the proof does not work in general) Silverman later uses this IV.4.4 in the proof of Theorem IV.6.1 and in Remark IV.7.1, where for Theorem IV.6.1 the assumption is that $R$ is a DVR and $p$ equals the characteristic of the residue field of $R$, and for Remark IV.7.1 the assumption is that $R$ is a ring of characteristic $p>0$. Is it valid to apply IV.4.4 to them under their respective assumptions?
My thoughts. I think when for IV.6.1 if $R=\mathbb{Z}_p$, then $p$ is a prime in $R$ and it is fine. For a general DVR $R$, if I know that the characteristic of its residue field $R/\mathfrak{m}$ is a prime (or $0$) in $R$, then it is fine in general. As for IV.7.1, where $p=\mathrm{char}R$, I think we need that $R$ is a domain so that $p=0$ is a ''prime'' in $R$.
This will become obvious if you write down $[p](T) = \sum_{n = 0}^\infty a_n T^n$ and then $[p]'(T) = \sum_{n = 0}^\infty na_nT^{n - 1}$.
What we have is that $na_n \in pR$ for each $n$. There are two cases: either $p\mid n$, or $p$ is prime to $n$.
In the second case, there exist $q, r \in \Bbb Z$ such that $pq + nr = 1$ and hence $a_n = pqa_n + r(na_n) \in pR$.