Exercise 40, § 1

Let $A$ be a ring. Let $\mathbf{J} = P \times Q$ be a decomposition of the monoid $J$ into a product of submonoids $P$ and $Q$ (which are therefore generated by the prime numbers they contain). Suppose that every $q \in Q$ is invertible in $A$, hence also in $U(A)$ (p. 52, Exercise 35, e)). a) For every $a \in U(A)$, the series

where $\mu$ denotes the Möbius function (Lie, II, p. 71), is convergent for the product topology of $U(A) = A^J$. Thus one defines the additive endomorphism $\varepsilon_Q = \sum_{q \in Q} \frac{\mu(q)}{q} V_q F_q$ of $U(A)$. For every $n \in \mathbf{J}$, the mapping $\Phi_n \circ \varepsilon_Q$ of $U(A)$ into $A$ is equal to 0 if $n \notin P$ and to $\Phi_n$ if $n \in P$. b) For every $q \in Q, q \neq 1$, one has (Use Exercise 36, c), p. 52.) c) Now prove that $\varepsilon_Q$ is an idempotent whose image is the intersection of the kernels of the $F_q$ ($q \in Q, q \neq 1$). For $m \in \mathbf{J}$ and $a \in A$, calculate $\varepsilon_Q V_m \tau(a)$. Deduce that the kernel of $\varepsilon_Q$ is the set of elements of the form $\sum_{n \in Q} V_n \tau(a_n)$, with $a_n \in A$. The subset $\varepsilon_Q U(A)$ of $U(A)$ is stable under addition and multiplication. Endowed with these two operations, it is a commutative ring, with unit $\varepsilon_Q(1_{U(A)})$, and the mapping $\varepsilon_Q : U(A) \to \varepsilon_Q U(A)$ is a ring homomorphism. d) For every $q \in Q$, let us put $e_q = \frac{1}{q} V_q \varepsilon_Q F_q$. Now prove that one has $e_q^2 = e_q$ and $e_q e_{q'} = 0$ if $q \neq q'$, and that, for every $a \in U(A)$ one has where the sum is convergent for the product topology of $U(A)$. (For () reduce to the case where $a = X \in U(R[X])$, where $R$ is the subring of $Q$ consisting of rational numbers with denominator in $Q$. It is then enough to apply $\Phi$ and to verify the analogue of () in $R[X]^J$, which results from the formula $\Phi_{pq} \circ e_{q'} = \Phi_{pq} \delta_{qq'}$ if $p \in P, q \in Q, q' \in Q$. The subset $e_q U(A)$ of $U(A)$ is stable under addition and multiplication. Endowed with these two operations, it is a commutative ring, with unit $e_q(1_{U(A)})$ and the mapping $e_q : U(A) \to e_q U(A)$ is a ring homomorphism. e) For every $q \in Q$, one has $F_q e_q = \varepsilon_Q F_q$ and $\left( \frac{1}{q} V_q \right) \varepsilon_Q = e_q \left( \frac{1}{q} V_q \right)$. Conclude that $a \mapsto \frac{1}{q} V_q(a)$ is an isomorphism of rings $e_q U(A) \to \varepsilon_Q U(A)$, with inverse $a \mapsto \frac{1}{q} V_q(a)$. f) Now prove that the canonical projection $\pi_P : U(A) \to U_P(A)$ (Exercise 39) defines an isomorphism of rings $\varepsilon_Q U(A) \to U_P(A)$. Deduce an isomorphism of rings of $U(A)$ onto $U_P(A)^Q$ transforming $a$ into $(\pi_P \varepsilon_Q F_q(a))_{q \in Q}$ for every $a \in U(A)$. g) If $A$ is a $\mathbf{Q}$-algebra, one can take $P = \{1\}$ and $Q = \mathbf{J}$. One deduces an isomorphism of rings $U(A) \to A^J$, which is none other than $\Phi$.

h) Suppose that $P = \{ p^n | n \in \mathbf{N} \}$, where $p$ is a prime number. Then every prime number $q \neq p$ is invertible in $A$. One obtains an isomorphism of rings $U(A) \to W(A)^Q$ (cf. exerc. 39, d)). We shall denote by $\omega_A$ the mapping of $W(A)$ into $U(A)$ which, by the isomorphism of $U(A)$ and $W(A)^Q$, corresponds to the inclusion of $W(A)$ in $W(A)^Q$ according to the first component.