topological module

A topological vector space is a topological module over a topological field.

An abelian topological group can be considered as a topological module over $\backslash Z,$ where $\backslash Z$ is the ring of integers with the discrete topology.

A topological ring is a topological module over each of its subrings.

A more complicated example is the $I$-adic topology on a ring and its modules. Let $I$ be an ideal of a ring $R.$ The sets of the form $x\; +\; I^n$ for all $x\; \backslash in\; R$ and all positive integers $n,$ form a base for a topology on $R$ that makes $R$ into a topological ring. Then for any left $R$-module $M,$ the sets of the form $x\; +\; I^n\; M,$ for all $x\; \backslash in\; M$ and all positive integers $n,$ form a base for a topology on $M$ that makes $M$ into a topological module over the topological ring $R.$

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• {{annotated link|Linear topology}}
• {{annotated link|Ordered topological vector space}}
• {{annotated link|Topological abelian group}}
• {{annotated link|Topological field}}
• {{annotated link|Topological group}}
• {{annotated link|Topological ring}}
• {{annotated link|Topological semigroup}}
• {{annotated link|Topological vector space}}

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• {{Cite book | last1=Atiyah | first1=Michael Francis | author1-link=Michael Atiyah | last2=MacDonald | first2=I.G. | author2-link=Ian G. Macdonald | title=Introduction to Commutative Algebra | publisher=Westview Press | isbn=978-0-201-40751-8 | year=1969}}
• {{Cite book|last=Kuz'min|first=L. V.|title=Encyclopedia of Mathematics|publisher=Kluwer Academic Publishers|year=1993|editor-last=Hazewinkel|editor-first=M.|volume=9|chapter=Topological modules}}

Category:Abstract algebra

Category:Topology

Category:Topological algebra

Category:Topological groups

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