Remark

Let $a = (a_n)_{n \in \mathbf{N}}$ and $u = (u_n)_{n \in \mathbf{N}}$ be elements of $A^\mathbf{N}$ such that $u = \Phi_A(a)$, and let $m$ be an integer $\geqslant 0$. From (10) one deduces the following assertions:

If the $u_n$, for $0 \leqslant n \leqslant m$, belong to a subring $B$ of $A$ and if, for every $x \in A$, the relation $p.x \in B$ implies $x \in B$, then the $a_n$, for $0 \leqslant n \leqslant m$, belong to $B$.

If $A$ is endowed with a graduation of type $\mathbf{N}$, if $p.1_A$ is not a divisor of 0 in $A$, if $d \in \mathbf{N}$ and if $u_n$ is homogeneous of degree $dp^n$ for $0 \leqslant n \leqslant m$, then $a_n$ is homogeneous of degree $dp^n$ for $0 \leqslant n \leqslant m$.