One can thus ask whether a given topological ring R is complete.

As may be expected, this produces a complete topological ring.

In the plane, split-complex numbers and dual numbers form alternative topological rings.

A topological ring is a topological module over each of its subrings.

The projective line over a ring for a topological ring may compactify it.

Endowed with the usual topology, the algebra of intervals forms a topological ring.

These examples of topological rings have the projective line as their one-point compactifications.

Every topological ring is a topological group (with respect to addition) and hence a uniform space in a natural manner.

This topological structure, together with the ring operations described above, form a topological ring.

This automatically gives R'X the structure of a topological ring (and even of a complete metric space).

Constructions of topological rings and modules (with V. I. Arnautov).

An approach to analysis based on topological groups, topological rings, and topological vector spaces.

When R is given the I-adic topology, R becomes a topological ring.

Every discrete valuation ring, being a local ring, carries a natural topology and is a topological ring.

The rational, real, complex and p-adic numbers are also topological rings (even topological fields, see below) with their standard topologies.

Seth Warner: Topological Rings.

This leads to concepts such as topological groups, topological vector spaces, topological rings and local fields.

Let A be a topological ring, and let B be a topological A-algebra.

In mathematics, a topological module is a module over a topological ring such that scalar multiplication and addition are continuous.

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.

In abstract algebra, a completion is any of several related functors on rings and modules that result in complete topological rings and modules.

Let A be a (Noetherian) topological ring, that is, a ring A which is a topological space such that the operations of addition and multiplication are continuous.

As for any topological ring, one can ask whether (R, m) is complete (as a topological space); if it is not, one considers its completion, again a local ring.

For example, n-by-n matrices over the real numbers could be given either the Euclidean topology, or the Zariski topology, and in either case one would obtain a topological ring.

Then the collection of all intervals [x,y] can be identified with the topological ring formed by the direct sum of R with itself where addition and multiplication are defined component-wise.