View source for Separable completion of a ring

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The completion of a [[topological ring]] $A/\bar{\mathfrak{o}}$, where $A$ is a topological ring and $\bar{\mathfrak{o}}$ is the closure in $A$ of the zero ideal $\mathfrak{o}$. The separable completion of a ring is also a topological ring and is usually denoted by $\hat A$. Every continuous homomorphism from $B$ into a complete separable ring $B$ can be uniquely extended to a continuous homomorphism $\hat A \rightarrow B$.

In the most important case where the ring $A$ has a [[linear topology]] defined by a fundamental system of ideals $\left(\mathfrak{a}_\lambda\right)_{\lambda \in \Lambda}$, the separable completion $\hat A$ is canonically identified with the [[projective limit]] $\lim_{\lambda \in \Lambda} A/\mathfrak{a}_\lambda$ of the discrete rings $A/\mathfrak{a}_\lambda$. The separable completion of a module is achieved in the same way.



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====References====
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Bourbaki,   "Algèbre commutative" , ''Eléments de mathématiques'' , Hermann  (1961)  pp. Chapt. 3. Graduations, filtrations, et topologies</TD></TR>
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