View source for Topological ring

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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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.

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

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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.

====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>