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A [[Ring|ring]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093110/t0931101.png" /> that is a [[Topological space|topological space]], such that the mappings
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{{MSC|13Jxx}}
 +
{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093110/t0931102.png" /></td> </tr></table>
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A topological ring is a [[Ring|ring]] $R$ that is a [[Topological space|topological space]], and such that the mappings
 
+
\[
are continuous. A topological ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093110/t0931103.png" /> is called separated if it is separated as a topological space (cf. [[Separation axiom|Separation axiom]]). In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093110/t0931104.png" /> is a [[Hausdorff space|Hausdorff space]]. Any subring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093110/t0931105.png" /> of a topological ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093110/t0931106.png" />, and also the quotient ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093110/t0931107.png" /> by an ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093110/t0931108.png" />, is a topological ring. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093110/t0931109.png" /> is separated and the ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093110/t09311010.png" /> is closed, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093110/t09311011.png" /> is a separated topological ring. The closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093110/t09311012.png" /> of a subring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093110/t09311013.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093110/t09311014.png" /> is also a topological ring. A direct product of topological rings is a topological ring in a natural way.
+
(x,y) \mapsto x-y
 
+
\]
A homomorphism of topological rings is a ring homomorphism which is also a continuous mapping. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093110/t09311015.png" /> is such a homomorphism, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093110/t09311016.png" /> is moreover an epimorphism and an open mapping, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093110/t09311017.png" /> is isomorphic as a topological ring to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093110/t09311018.png" />. Banach algebras are an example of topological rings. An important type of topological ring is defined by the property that it has a fundamental system of neighbourhoods of zero consisting of some set of ideals. For example, to any ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093110/t09311019.png" /> in a commutative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093110/t09311020.png" /> one can associate the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093110/t09311022.png" />-adic topology, in which the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093110/t09311023.png" /> for all natural numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093110/t09311024.png" /> form a fundamental system of neighbourhoods of zero. This topology is separated if the condition
+
and
 
+
\[
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093110/t09311025.png" /></td> </tr></table>
+
(x,y) \mapsto xy
 +
\]
 +
are continuous. A topological ring $R$ is called separated if it is separated as a topological space (cf. [[Separation axiom|Separation     axiom]]). In this case $R$ is a [[Hausdorff space|Hausdorff     space]]. Any subring $M$ of a topological ring $R$, and also the quotient ring $R/J$ by an ideal $J$, is a topological ring. If $R$ is separated and the ideal $J$ is closed, then $R/J$ is a separated topological ring. The closure $\bar{M}$ of a subring $M$ in $R$ is also a topological ring. A direct product of topological rings is a topological ring in a natural way.
  
 +
A homomorphism of topological rings is a ring homomorphism which is also a continuous mapping. If $f:R_1 \rightarrow R_2$ is such a homomorphism, where $f$ is moreover an epimorphism and an open mapping, then $R_2$ is isomorphic as a topological ring to $R_1/\mathrm{Ker}f$. Banach algebras are an example of topological rings. An important type of topological ring is defined by the property that it has a fundamental system of neighbourhoods of zero consisting of some set of ideals. For example, to any ideal $\mathfrak{m}$ in a commutative ring $R$ one can associate the $\mathfrak{m}$-adic topology, in which the sets $\mathfrak{m}^n$ for all natural numbers $n$ form a fundamental system of neighbourhoods of zero. This topology is separated if the condition
 +
\[
 +
\bigcap_n \,\mathfrak{m}^n = 0
 +
\]
 
is satisfied.
 
is satisfied.
  
For a topological ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093110/t09311026.png" /> one can define its completion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093110/t09311027.png" />, which is a complete topological ring, and a separated topological ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093110/t09311028.png" /> can be imbedded as an everywhere-dense subset in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093110/t09311029.png" />, which is also separated in this case. The additive group of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093110/t09311030.png" /> coincides with the completion of the additive group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093110/t09311031.png" />, as an Abelian topological group.
+
For a topological ring $R$ one can define its completion $\hat{R}$, which is a complete topological ring, and a separated topological ring $R$ can be imbedded as an everywhere-dense subset in $\hat{R}$, which is also separated in this case. The additive group of the ring $\hat{R}$ coincides with the completion of the additive group of $R$, as an Abelian topological group.
  
====References====
+
====References====  
<table><TR><TD valign="top">[1]</TD> <TD valign="top"N. Bourbaki,   "Elements of mathematics. General topology" , Addison-Wesley (1966) (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"N. Bourbaki,   "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"L.S. Pontryagin,   "Topological groups" , Princeton Univ. Press (1958) (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"B.L. van der Waerden,   "Algebra" , '''1–2''' , Springer (1967–1971) (Translated from German)</TD></TR></table>
+
{|
 +
|-
 +
|valign="top"|{{Ref|Bo}}||valign="top"| N. Bourbaki, "Elements of mathematics. General topology", Addison-Wesley (1966) (Translated from French)
 +
|-
 +
|valign="top"|{{Ref|Bo2}}||valign="top"| N. Bourbaki, "Elements of mathematics. Commutative algebra", Addison-Wesley (1972) (Translated from French)
 +
|-
 +
|valign="top"|{{Ref|Po}}||valign="top"| L.S. Pontryagin, "Topological groups", Princeton Univ. Press (1958) (Translated from Russian)
 +
|-
 +
|valign="top"|{{Ref|Wa}}||valign="top"| B.L. van der Waerden, "Algebra", '''1–2''', Springer (1967–1971) (Translated from German)
 +
|-
 +
|}

Latest revision as of 23:45, 23 July 2012

2010 Mathematics Subject Classification: Primary: 13Jxx [MSN][ZBL]

A topological ring is a ring $R$ that is a topological space, and such that the mappings \[ (x,y) \mapsto x-y \] and \[ (x,y) \mapsto xy \] are continuous. A topological ring $R$ is called separated if it is separated as a topological space (cf. Separation axiom). In this case $R$ is a Hausdorff space. Any subring $M$ of a topological ring $R$, and also the quotient ring $R/J$ by an ideal $J$, is a topological ring. If $R$ is separated and the ideal $J$ is closed, then $R/J$ is a separated topological ring. The closure $\bar{M}$ of a subring $M$ in $R$ is also a topological ring. A direct product of topological rings is a topological ring in a natural way.

A homomorphism of topological rings is a ring homomorphism which is also a continuous mapping. If $f:R_1 \rightarrow R_2$ is such a homomorphism, where $f$ is moreover an epimorphism and an open mapping, then $R_2$ is isomorphic as a topological ring to $R_1/\mathrm{Ker}f$. Banach algebras are an example of topological rings. An important type of topological ring is defined by the property that it has a fundamental system of neighbourhoods of zero consisting of some set of ideals. For example, to any ideal $\mathfrak{m}$ in a commutative ring $R$ one can associate the $\mathfrak{m}$-adic topology, in which the sets $\mathfrak{m}^n$ for all natural numbers $n$ form a fundamental system of neighbourhoods of zero. This topology is separated if the condition \[ \bigcap_n \,\mathfrak{m}^n = 0 \] is satisfied.

For a topological ring $R$ one can define its completion $\hat{R}$, which is a complete topological ring, and a separated topological ring $R$ can be imbedded as an everywhere-dense subset in $\hat{R}$, which is also separated in this case. The additive group of the ring $\hat{R}$ coincides with the completion of the additive group of $R$, as an Abelian topological group.

References

[Bo] N. Bourbaki, "Elements of mathematics. General topology", Addison-Wesley (1966) (Translated from French)
[Bo2] N. Bourbaki, "Elements of mathematics. Commutative algebra", Addison-Wesley (1972) (Translated from French)
[Po] L.S. Pontryagin, "Topological groups", Princeton Univ. Press (1958) (Translated from Russian)
[Wa] B.L. van der Waerden, "Algebra", 1–2, Springer (1967–1971) (Translated from German)
How to Cite This Entry:
Topological ring. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Topological_ring&oldid=17173
This article was adapted from an original article by L.V. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article