Proposition 6

A Hausdorff topological ring $A$ is isomorphic to a dense subring of a complete Hausdorff ring $\hat{A}$, which is determined up to isomorphism (and is called the completion of $A$).

If $A$ is commutative (resp. has an identity element) then the same is true of $\hat{A}$ (principle of extension of identities).

Let $A$ be a topological ring, not necessarily Hausdorff; let $N$ be the closure of $\{0\}$ in $A$, and let $A' = A/N$ be the Hausdorff ring associated with $A$. Then $\hat{A}'$, the completion of $A'$, is called the Hausdorff completion of $A$ and is also denoted by $\hat{A}$. One shows as in § 3, no. 4, Proposition 8 that every continuous ring homomorphism $u$ of $A$ with a complete Hausdorff topological ring $C$ can be uniquely factorized as $u = v \circ \varphi$, where $v$ is a continuous homomorphism of $\hat{A}$ into $C$ and $\varphi$ is the canonical mapping of $A$ into $\hat{A}$. If $A, B$ are two topological rings, and $u : A \to B$ is a continuous homomorphism, there is therefore a unique continuous homomorphism $\hat{u} : \hat{A} \to \hat{B}$ such that the diagram

is commutative ($\varphi$ and $\psi$ being the canonical mappings); we have only to apply the preceding result to $\psi \circ u$.