Theorem 1

a) Let A be a (commutative) ring. Endowed with the addition $S_A$ and the multiplication $P_A$, W(A) is a (commutative) ring. The identity element for the addition is the sequence $0_A$ all of whose terms are zero; the identity element for the multiplication is the sequence $1_A$ all of whose terms are zero except the one of index 0 which is equal to $1_A$. The opposite of an element a of W(A) is $I_A(a)$.

b) Let $\rho : B \to A$ be a homomorphism of rings. Then $W(\rho) : W(B) \to W(A)$ is a homomorphism of rings.

c) Let $A$ be a ring. The mapping $\Phi_A$ is a homomorphism of rings from $W(A)$ into the product ring $A^\mathbf{N}$. In particular, for all $n \in \mathbf{N}$, the mapping $\Phi_n : a \mapsto \Phi_n(a_0, ..., a_n)$ is a homomorphism of rings from $W(A)$ into $A$.

Taking into account formulas (16), (17), (19) and (20), it suffices to prove assertion $a$.

Let $\rho : B \to A$ be a homomorphism of rings satisfying the conditions of Lemma 3. Let $B'$ be the subring of $B^\mathbf{N}$ formed by the elements $(b_n)_{n \in \mathbf{N}}$ such that $\sigma(b_n) \equiv b_{n+1} \mod. p^{n+1}$.B for all $n \in \mathbf{N}$. According to Prop. 2 of No. 2, $\Phi_B$ induces a bijection $\Phi'_B$ from $W(B)$ onto $B'$. Viewed from formulas (16) to (18) and the relations $\Phi_n(0_B) = 0$ and $\Phi_n(1_B) = 1_B$ ($n \in \mathbf{N}$), one sees by transport of structure that $W(B)$ is a ring, with identity element $0_B$ for the addition, $1_B$ for the multiplication, the opposite of $b$ being $I_B(b)$.

The mapping $W(\rho) : W(B) \to W(A)$ is surjective. According to formulas (19) and (20), the equivalence relation R on $W(B)$ associated with the mapping $W(\rho)$ is compatible with the ring structure of $W(B)$. Since $W(\rho)$ induces a bijection $\Psi$ from the quotient ring $W(B)/R$ onto $W(A)$, compatible with the laws of addition and multiplication, assertion $a$ follows from this by transport of structure.