Exercise 41, § 1

Let $A$ be a ring. a) Let $\mu : A \to U(A)$ be a ring homomorphism such that $\Phi_1 \circ \mu = \mathrm{Id}_A$. Put $\sigma = \Phi \circ \mu$ and $\sigma(a) = (\sigma_n(a))_{n \in J}$ for $a \in A$. The mappings $\sigma_n : A \to A$ ($n \in J$) satisfy the following conditions: (1) $\sigma_n$ is a ring homomorphism for every $n \in J$, and $\sigma_1 = \mathrm{Id}_A$. (2) For every prime number $p$ and for every $a \in A$, one has $\sigma_p(a) \equiv a^p \mod. pA$ and

b) Suppose that the additive group of $A$ has no $\mathbf{Z}$-torsion. Let $\sigma = (\sigma_n)_{n \in J}$ be a family of mappings $\sigma_n : A \to A$ satisfying conditions (1) and (2) of $a$. There exists a unique ring homomorphism $\mu : A \to U(A)$ such that $\Phi_1 \circ \mu = \mathrm{Id}_A$ and $\Phi \circ \mu = \sigma$. (Apply Exercises 32 and 33, d), p. 51.) c) Suppose that the additive group of $A$ has no $\mathbf{Z}$-torsion. There exists a unique ring homomorphism such that $\Phi^{U(A)} \circ \mu^A$ is the mapping defined by $F^A(a) = (F_n(a))_{n \in J}$. (Reduce to the case where $A = \mathbf{Z}[X]$. Apply $b$ and Exercises 36 and 38, b), p. 52 and 53.) d) Let $X = (X_n)_{n \in J}$ be a family of indeterminates considered as an element of $U(\mathbf{Z}[X])$. Put where $\mu_{n,m}(X) \in \mathbf{Z}[X]$ for $n, m$ in $J$. For every ring $A$ define $\mu^A : U(A) \to U(U(A))$ by $\mu^A(a) = ((\mu_{n,m}(a))_{m \in J})_{n \in J}$. Now prove that $\mu^A$ is a ring homomorphism such that the following diagram is commutative

(1)

Here by definition, $f^A(a) = (f_n(a))_{n \in J}$ for $a \in A^J$, and $\Phi^A$ (resp. $\Phi^{U(A)}$) is the homomorphism $\Phi$ associated with the ring $A$ (resp. $U(A)$). e) For every ring homomorphism $\rho : B \to A$ the diagram

is commutative.

f) Now prove that the following diagram is commutative

(Reduce to the case where $A = \mathbf{Q}[X]$. Then $A$ is a $\mathbf{Q}$-algebra, hence $\Phi^A : U(A) \to A^J$ is an isomorphism and $U(A)$ is also a $\mathbf{Q}$-algebra. If, with the aid of diagram (1) and by transport of structure, one replaces $\mu^A$ by $f^A$ when $A$ is a $\mathbf{Q}$-algebra, one is reduced to proving the commutativity of the diagram

One has

and

g) For all $a \in A$, one has

h) One has the following commutative diagram

where the mapping $s_A$ is that defined in Exercise 15, p. 44, where the mapping $\varphi$ from $U(A)$ into $W(A)$ is obtained by identifying $W(A)$ and $U_{p^\infty}(A)$ (p. 53, Exercise 39), and the mapping $\psi$ is the composite of $U(\varphi)$ and the mapping from $U(W(A))$ into $W(W(A))$ obtained by identifying $W(W(A))$ with $U_{p^\infty}(W(A))$.)