Exercise 2, § 3

a) Let $A$ be an integral domain, union of an increasing filtered family $(A_\lambda)$ of subrings. Suppose that each of the $A_\lambda$ is factorial and that if $\lambda \leq \mu$, every extremal element of $A_\lambda$ is extremal in $A_\mu$. Now prove that $A$ is a factorial ring whose set of extremal elements is the union of the sets of extremal elements of each of the $A_\lambda$ (cf. § 2, exerc. 19 e)).

b) Deduce from a) that for every factorial ring $A$, the polynomial ring $A[X_\lambda]_{\lambda \in L}$ with respect to any family of indeterminates is factorial.

c) Let $A$ be a factorial ring such that every formal power series ring $A[[X_1, \ldots, X_n]]$ with respect to a finite number of indeterminates is factorial. Now prove that every formal power series ring $A[[X_\lambda]]_{\lambda \in L}$ with respect to any family of indeterminates ($Alg.$, chap. IV, § 5, exerc. 1) is factorial (same method).