Exercise 2, § 3

(a) Let $\mathbf{A}$ be an integral domain, the union of a right directed family $(\mathbf{A}_\lambda)$ of subrings. Suppose that each of the $\mathbf{A}_\lambda$ is factorial and that, if $\lambda \leq \mu$, every extremal element of $\mathbf{A}_\lambda$ is extremal in $\mathbf{A}_\mu$. Show that $\mathbf{A}$ is a factorial domain whose set of extremal elements is the union of the sets of extremal elements of each of the $\mathbf{A}_\lambda$ (cf. § 2, Exercise 19 (e)).

(b) Deduce from (a) that for every factorial domain $\mathbf{A}$ the polynomial domain $\mathbf{A}[X_\lambda]_{\lambda \in L}$ in any family of indeterminates is factorial.

(c) Let $\mathbf{A}$ be a factorial domain such that every domain of formal power series $\mathbf{A}[[X_1, \ldots, X_n]]$ in a finite number of indeterminates is factorial. Show that every domain of formal power series $\mathbf{A}[[X_\lambda]]_{\lambda \in L}$ in any family of indeterminates (Algebra, Chapter IV, § 5, Exercise 1) is factorial (same method).