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.



[a1] N. Bourbaki, "Algèbre commutative" , Eléments de mathématiques , Hermann (1961) pp. Chapt. 3. Graduations, filtrations, et topologies
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Separable completion of a ring. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article