nLab
topological ring

Context

Topology

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Basic concepts

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Basic statements

Theorems

Analysis Theorems

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Higher algebra

Algebraic theories

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Algebras and modules

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Higher algebras

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Model category presentations

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Geometry on formal duals of algebras

Theorems

Contents

Definition

A topological ring is a ring internal to Top, a ring object in Top:

a topological space RR equipped with the structure of a ring on its underlying set, such that addition ++ and multiplication \cdot are continuous functions.

Remarks:

  • The structure of a topological ring (R,+,)(R,+,\cdot) makes RR a uniform space.

  • It is automatic that negation in a topological unital ring is continuous, since it is the operation of multiplication with 1-1, and multiplication is continuous in each argument. Hence for (R,+,)(R,+,\cdot) a topological ring, then (R,+)(R,+) is a topological group.

  • A topological field is a topological ring KK whose underlying ring is in fact a field and such that reciprocation () 1:K{0}K{0}(-)^{-1}: K \setminus \{0\} \to K \setminus \{0\} is continuous. This latter condition is the same as demanding that the subspace topology on K{0}K \setminus \{0\} induced by the embedding K{0}KK \setminus \{0\} \hookrightarrow K coincide with the subspace topology induced by the embedding K{0}K×K:x(x,x 1)K \setminus \{0\} \to K \times K: x \mapsto (x, x^{-1}). More at topological field.

Remark

In a topological ring, the closure of {0}\{0\} is an ideal. It follows that for a topological field FF, either 00 is a closed point (so that FF is T 1T_1 and therefore completely regular Hausdorff, by standard arguments in the theory of uniform spaces), or is a codiscrete space.

A topological algebra over a topological ring RR is a topological ring SS together with a topological ring map RSR \to S that makes SS an RR-algebra at the underlying set level (a topological associative algebra).

Examples

References

Lecture notes include

See also

Last revised on May 30, 2017 at 05:42:30. See the history of this page for a list of all contributions to it.