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Topological module

From Wikipedia, the free encyclopedia

In mathematics, a topological module is a module over a topological ring such that scalar multiplication and addition are continuous.

Examples

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

An abelian topological group can be considered as a topological module over where 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 -adic topology on a ring and its modules. Let be an ideal of a ring The sets of the form for all and all positive integers form a base for a topology on that makes into a topological ring. Then for any left -module the sets of the form for all and all positive integers form a base for a topology on that makes into a topological module over the topological ring

See also

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References

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  • Atiyah, Michael Francis; MacDonald, I.G. (1969). Introduction to Commutative Algebra. Westview Press. ISBN 978-0-201-40751-8.
  • Kuz'min, L. V. (1993). "Topological modules". In Hazewinkel, M. (ed.). Encyclopedia of Mathematics. Vol. 9. Kluwer Academic Publishers.