# Topological ring

In mathematics, a **topological ring** is a ring *R* that is also a topological space such that both the addition and the multiplication are continuous as maps:^{[1]}

Topological rings are fundamentally related to topological fields and arise naturally while studying them, since for example completion of a topological field may be a topological ring which is not a field.^{[2]}

The group of units *R*^{×} of a topological ring *R* is a topological group when endowed with the topology coming from the embedding of *R*^{×} into the product *R* × *R* as (*x*,*x*^{−1}). However, if the unit group is endowed with the subspace topology as a subspace of *R*, it may not be a topological group, because inversion on *R*^{×} 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 *R*^{×} is continuous in the subspace topology of *R* then these two topologies on *R*^{×} 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 +) in which multiplication is continuous, too.

Topological rings occur in mathematical analysis, for example as rings of continuous real-valued functions on some topological space (where the topology is given by pointwise convergence), or as rings of continuous linear operators on some normed vector space; all Banach algebras are topological rings. The rational, real, complex and *p*-adic numbers are also topological rings (even topological fields, see below) with their standard topologies. In the plane, split-complex numbers and dual numbers form alternative topological rings. See hypercomplex numbers for other low-dimensional examples.

In algebra, the following construction is common: one starts with a commutative ring *R* containing an ideal *I*, and then considers the ** I-adic topology** on

*R*: a subset

*U*of

*R*is open if and only if for every

*x*in

*U*there exists a natural number

*n*such that

*x*+

*I*

^{n}⊆

*U*. This turns

*R*into a topological ring. The

*I*-adic topology is Hausdorff if and only if the intersection of all powers of

*I*is the zero ideal (0).

The *p*-adic topology on the integers is an example of an *I*-adic topology (with *I* = (*p*)).

The rings of formal power series and the *p*-adic integers are most naturally defined as completions of certain topological rings carrying *I*-adic topologies.

Some of the most important examples are topological fields. A topological field is a topological ring that is also a field, and such that inversion of non zero elements is a continuous function. The most common examples are the complex numbers and all its subfields, and the valued fields, which include the p-adic fields.