Exercise 28, § 1

Let $J$ be the set of integers $\geqslant 1$. For every $n \in J$, let us denote by $J_n$ the set of integers $d \geqslant 1$ which divide $n$. Let $A$ be a ring. We make $A^J$ into a product ring. We define the mappings $\Phi, f_n, v_n (n \in J)$ of $A^J$ into itself by the following formulae. For $a = (a_j)_{j \in J}$, put

it being understood that $a_{m/n} = 0$ if $m/n \notin J$. Note that $\Phi_n$ depends only on the $a_d$ ($d \in J_n$). We shall sometimes write $\Phi_n((a_d)_{d \in J_n})$ in place of $\Phi_n(a)$.

For every prime number $p$, we denote by $w_p$ the $p$-adic valuation of $\mathbf{Q}$. These notations will be retained in the exercises that follow in § 1. a) $f_n$ is an endomorphism of the ring $A^J$, and $f_n \circ f_m = f_{nm}$ for all $n, m$ in $J$. b) $v_n$ is an endomorphism of the additive group of $A^J$, and $v_n \circ v_m = v_{nm}$ for all $n, m$ in $J$. Let $n, m$ be in $J$ and $d = \mathrm{pgcd}(m, n)$ c) One has $f_n \circ v_m = d \cdot v_{m/d} \circ f_{n/d}$. In particular $f_n \circ v_n = n.\mathrm{Id}$, and $f_n \circ v_m = v_m \circ f_n$ if $d = 1$. d) For all $a, b$ in $A^J$ one has

(The second formula follows from the first, and the third from the second by taking $a = 1$.) In particular, one has

and

if $d = 1$.