Exercise 52, § 1

Let $(A_i)_{i \in I}$ be a family of rings, $A = \prod_i A_i$, and $p_i : A \to A_i$ the canonical projection for each $i \in I$. a) The mapping $\alpha : A[[T]] \to \prod_i A_i[[T]]$, which maps $\sum a_n T^n$ to $(\sum p_i(a_n) T^n)_{i \in I}$, is an isomorphism of rings. By restriction, it defines an isomorphism of rings $\alpha : \Lambda(A) \to \prod_i \Lambda(A_i)$. Moreover, for every $n \in J$, one has $\alpha \circ \lambda_n^A = (\prod_i \lambda_n^{A_i}) \circ \alpha$. (Use the "universal polynomials" of Exercise 46, d), p. 59.) b) Suppose that, for each $i \in I$, $A_i$ is endowed with a pre-$\lambda$-ring structure $\lambda_i : A_i \to \Lambda(A_i)$. By identifying $\Lambda(A)$ with $\prod_i \Lambda(A_i)$ by means of $\alpha$, one defines $\lambda = \prod_i \lambda_i : A \to \Lambda(A)$. Then $(A, \lambda)$ is a pre-$\lambda$-ring, called the product of the family $((A_i, \lambda_i))_{i \in I}$. For $(A, \lambda)$ to be a $\lambda$-ring, it is necessary and sufficient that $(A_i, \lambda_i)$ be one for each $i \in I$.