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Difference between revisions of "Separable completion of a ring"

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The completion of the [[Topological ring|topological ring]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084460/s0844601.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084460/s0844602.png" /> is a topological ring and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084460/s0844603.png" /> is the closure in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084460/s0844604.png" /> of the zero ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084460/s0844605.png" />. The separable completion of a ring is also a topological ring and is usually denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084460/s0844606.png" />. Every continuous homomorphism from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084460/s0844607.png" /> into a complete separable ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084460/s0844608.png" /> can be uniquely extended to a continuous homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084460/s0844609.png" />.
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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 topology of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084460/s08446010.png" /> is linear and is defined by a fundamental system of ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084460/s08446011.png" />, the separable completion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084460/s08446012.png" /> is canonically identified with the [[Projective limit|projective limit]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084460/s08446013.png" /> of the discrete rings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084460/s08446014.png" />. The separable completion of a module is achieved in the same way.
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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====
 
====References====
<table><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></table>
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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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Latest revision as of 15:55, 21 December 2014

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.


Comments

References

[a1] N. Bourbaki, "Algèbre commutative" , Eléments de mathématiques , Hermann (1961) pp. Chapt. 3. Graduations, filtrations, et topologies
How to Cite This Entry:
Separable completion of a ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Separable_completion_of_a_ring&oldid=35773
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article