Editing Topological ring (section)

==Examples==

Topological rings occur in [[mathematical analysis]], for example as rings of continuous real-valued [[Function (mathematics)|function]]s on some topological space (where the topology is given by pointwise convergence), or as rings of continuous [[linear operator]]s on some [[normed vector space]]; all [[Banach algebra]]s are topological rings. The [[Rational number|rational]], [[Real number|real]], [[Complex number|complex]] and [[p-adic number|<math>p</math>-adic]] numbers are also topological rings (even topological fields, see below) with their standard topologies. In the plane, [[split-complex number]]s and [[dual numbers]] form alternative topological rings. See [[hypercomplex numbers]] for other low-dimensional examples.

In [[commutative algebra]], the following construction is common: given an [[Ideal (ring)|ideal]] <math>I</math> in a [[commutative]] ring <math>R,</math> the [[Adic topology|{{mvar|I}}-adic topology]] on <math>R</math> is defined as follows: a [[subset]] <math>U</math> of <math>R</math> is open [[if and only if]] for every <math>x \in U</math> there exists a natural number <math>n</math> such that <math>x + I^n \subseteq U.</math> This turns <math>R</math> into a topological ring.  The <math>I</math>-adic topology is [[Hausdorff space|Hausdorff]] if and only if the [[Intersection (set theory)|intersection]] of all powers of <math>I</math> is the zero ideal <math>(0).</math> 

The <math>p</math>-adic topology on the [[integer]]s is an example of an <math>I</math>-adic topology (with <math>I = p\Z</math>).