Corollary

Let $A$ be a principal ideal domain and $P$ a representative system of extremal elements of $A$ (Algebra, Chapter VII, § 1, no. 3). The topology on $A$ with respect to which the ideals $\neq 0$ of $\mathbf{A}$ form a fundamental system of neighbourhoods of 0, which is compatible with the ring structure on $\mathbf{A}$, is Hausdorff and the completion of $\mathbf{A}$ with this topology is canonically isomorphic to the product of the completions of $\mathbf{A}$ with respect to the $x$-adic topologies, where $x$ runs through $P$.

The principal ideals $(x)$ where $\pi \in P$ are maximal and distinct and hence relatively prime, we have already seen (no. 12, Example 3) that the $x$-adic topologies are Hausdorff and hence so is the topology defined in the statement of Proposition 17, which is finer than each of the $x$-adic topologies.

If the Corollary to Proposition 17 is applied when $\mathbf{A} = \mathbf{Z}$, we denote by $\hat{\mathbf{Z}}$ the completion of $\mathbf{Z}$ with respect to the topology for which all the ideals $\neq 0$ of $\mathbf{Z}$ form a fundamental system of neighbourhoods of 0, the ring isomorphic to the product $\prod_{p \in P} \mathbf{Z}_p$ of the rings of $p$-adic integers (P being the set of prime numbers).

Remarks

(1) Clearly, under the conditions of Proposition 17, the topology $\mathscr{T}$ is the least upper bound of the $m_\lambda$-adic topologies on $\mathbf{A}$.

(2) Every closed ideal $a$ of $\prod_{\lambda \in L} \hat{A}_\lambda$ is identical with the product of its projections $a_\lambda = \mathrm{pr}_\lambda(a)$, which are closed ideals in the $A_\lambda$; for $\hat{A}_\lambda$ is canonically identified with a closed ideal $A'_\lambda$ of $\prod_{\lambda} \hat{A}_\lambda$ and $a_\lambda$ with $a \cap A'_\lambda$ (Algebra, Chapter I, §8, no. 10, Proposition 6), the sum of the $a_\lambda$ is dense in the product $\prod_\lambda a_\lambda$ (General Topology, Chapter III, §2, no. 9, Proposition 25) and the latter is closed in $\prod_\lambda \hat{A}_\lambda$, whence our assertion.