topological ring
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×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.
| 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 |