Editing Topological ring (section)

==General comments==

The [[group of units]] <math>R^\times</math> of a topological ring <math>R</math> is a [[topological group]] when endowed with the topology coming from the [[Embedding#General topology|embedding]] of <math>R^\times</math> into the product <math>R \times R</math> as <math>\left(x, x^{-1}\right).</math> However, if the unit group is endowed with the [[subspace topology]] as a subspace of <math>R,</math> it may not be a topological group, because inversion on <math>R^\times</math> need not be continuous with respect to the subspace topology. An example of this situation is the [[adele ring]] of a [[global field]]; its unit group, called the [[idele group]], is not a topological group in the subspace topology. If inversion on <math>R^\times</math> is continuous in the subspace topology of <math>R</math> then these two topologies on <math>R^\times</math> are the same.

If one does not require a ring to have a unit, then one has to add the requirement of continuity of the additive inverse, or equivalently, to define the topological ring as a ring that is a [[topological group]] (for <math>+</math>) in which multiplication is continuous, too.