Proposition 5

Let $H$ be a dense subring of a topological ring $A$, and let $K$ be a subring (resp. left ideal, right ideal, two-sided ideal) of $H$. Then the closure $\overline{K}$ of $K$ in $A$ is a subring (resp. left ideal, right ideal, two-sided ideal) of $A$.

The proof is the same as for Proposition 8 of § 2, n. 3 : if for example $K$ is a left ideal in $H$, then the mapping $(z, x) \to zx$ is continuous on $A \times A$ and maps $H \times K$ into $K$; hence it maps $A \times K = \overline{H} \times K$ into $\overline{H}$.

Let $H$ be a two-sided ideal in a topological ring $A$. By the same argument as for quotient groups, we see that the quotient of the topology of $A$ by the relation $x - y \in H$ is compatible with the ring structure of $A/H$. In particular, if $A$ is not Hausdorff, then the closure $N$ of $\{0\}$ in $A$ is a closed two-sided ideal, by Proposition 5; the quotient ring, which is Hausdorff ($\S$ 2, no. 5, Proposition 13) is called the Hausdorff ring associated with $A$.

Let $(A_i)_{i \in I}$ be a family of topological rings. On the set $A = \prod_{i \in I} A_i$, the product of the topologies of the $A_i$ is compatible with the product of the ring structures of the $A_i$ (same proof as for product groups); the topological ring $A$ thus defined is called the product of the topological rings $A_i$.