topological ring

A ring R which is a topological spaceMathworldPlanetmath is called a topological ring if the addition, multiplication, and the additive inverse functions are continuous functionsMathworldPlanetmath from R×R to R.

A topological division ring is a topological ring such that the multiplicative inverseMathworldPlanetmath 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.

Title topological ring
Canonical name TopologicalRing
Date of creation 2013-03-22 12:45:59
Last modified on 2013-03-22 12:45:59
Owner djao (24)
Last modified by djao (24)
Numerical id 6
Author djao (24)
Entry type Definition
Classification msc 12J99
Classification msc 13J99
Classification msc 54H13
Related topic TopologicalGroup
Related topic TopologicalVectorSpace
Defines topological field
Defines topological division ring