Remark

Let us take up again the notations of the remark of No. 3. Let $A$ be a ring. If $a$ and $b$ are elements of $A^J$ and $r = (r_j)_{j \in J}$ is an element of $A^J$, we denote by $r_A(a, b)$ the element $(r_j(a, b))_{j \in J}$ of $A^J$. Let us denote by $U(A)$ the set $A^J$ endowed with the composition laws $s_A$ and $p_A$. One can show (p. 52, Exercise 35) that, endowed with the addition $s_A$ and the multiplication $p_A$, $U(A)$ is a (commutative) ring; it is called the universal Witt ring of $A$. The identity element for the addition is the element of $U(A)$ all of whose components are zero; the identity element for the multiplication is the element of $U(A)$ all of whose components are zero except that of index 1 which is equal to $1_A$; the opposite of an element $a$ of $U(A)$ is $i_A(a)$. The mapping $\varphi_A$ is a ring homomorphism of $U(A)$ into the product ring $A^J$.

Let $\rho : B \to A$ be a ring homomorphism; we denote by $U(\rho)$ the mapping from $B^J$ into $A^J$ which associates with the element $(b_j)_{j \in J}$ of $B^J$ the element $(\rho(b_j))_{j \in J}$ of $A^J$. One can show (loc. cit.) that $U(\rho)$ is a ring homomorphism of $U(B)$ into $U(A)$.