Exercise 36, § 1

Let $A$ be a ring. Let $n, m$ be in $J$ and let $d = \operatorname{pgcd}(n, m)$. a) The mapping $a \mapsto F_n(a) = (F_{n,r}(a))_{r \in J}$ is an endomorphism of the ring U(A), and one has $F_n \circ F_m = F_{mn}$, $\Phi_n \circ F_m = \Phi_{nm}$. b) The mapping $a \mapsto V_n(a) = (V_{n,r}(a))_{r \in J}$ is an endomorphism of the additive group of U(A), and one has $V_n \circ V_m = V_{nm}$. Moreover, $\Phi_n \circ V_q$ is equal to 0 if $q$ does not divide $n$ and to $q \Phi_{n/q}$ if $q$ divides $n$. c) One has $F_n \circ V_m = d \times V_{m/d} \circ F_{n/d}$; in other words $F_n(V_m(a))$ (for $a \in U(A)$) is the sum in U(A) of $d$ terms equal to $V_{m/d}(F_{n/d}(a))$. In particular, one has $F_n(V_n(a)) = n \times a$ and $F_n \circ V_m = V_m \circ F_n$ if $d = 1$. d) For any $a, b$ in U(A), one has

In particular, one has and if $d = 1$. (Reduce to the case where $A = \mathbf{Z}[X, Y]$, $a = X$, $b = Y$ and use Exercise 33, d), p. 51 and Exercise 28, p. 50.)

EXERCISES