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Svetlana Velmar-Janković
In mathematics, a topological module is a module over a topological ring such that scalar multiplication and addition are continuous.
Examples
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
- Linear topology
- Ordered topological vector space
- Topological abelian group – topological group whose group is abelian
- Topological field – Algebraic structure with addition, multiplication, and division
- Topological group – Group that is a topological space with continuous group action
- Topological ring – ring where ring operations are continuous
- Topological semigroup – semigroup with continuous operation
- Topological vector space – Vector space with a notion of nearness
References
- 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.