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topological ring (Definition)

A ring $ R$ which is a topological space is called a topological ring if the addition, multiplication, and the additive inverse functions are continuous functions from $ R \times R$ to $ R$.

A topological division ring is a topological ring such that the multiplicative inverse function is continuous away from 0. A topological field is a topological division ring that is a field.

Remark. It is easy to see that if $ R$ contains the multiplicative identity $ 1$, then $ R$ is a topological ring iff addition and multiplication are continuous. This is true because the additive inverse of an element can be written as the product of the element and $ -1$. However, if $ R$ does not contain $ 1$, it is necessary to impose the continuity condition on the additive inverse operation.



"topological ring" is owned by djao. [ full author list (2) ]
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See Also: topological group (obsolete), topological vector space

Also defines:  topological field, topological division ring
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Cross-references: operation, necessary, product, inverse, iff, multiplicative identity, contains, easy to see, field, function, multiplicative inverse, continuous functions, inverse functions, additive, multiplication, addition, topological space, ring
There are 13 references to this entry.

This is version 3 of topological ring, born on 2002-06-08, modified 2008-06-28.
Object id is 3076, canonical name is TopologicalRing.
Accessed 4952 times total.

Classification:
AMS MSC: 13J99 (Commutative rings and algebras :: Topological rings and modules :: Miscellaneous)
 12J99 (Field theory and polynomials :: Topological fields :: Miscellaneous)
 54H13 (General topology :: Connections with other structures, applications :: Topological fields, rings, etc.)
Pending Errata and Addenda
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Discussion
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topological field by igor on 2002-06-09 00:03:22
Is it not necessary for a topological field
to have a continuous inverse operation, just
like for a topological group?

Or does this already follow from the facts
that it is a field and a topological ring?
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