- Topological ring
In

mathematics , a**topological ring**is a ring "R" which is also atopological space such that both the addition and the multiplication are continuous as maps:"R" × "R" → "R",

where "R" × "R" carries the

product topology .**General comments**The

group of units of "R" may not be atopological group using thesubspace topology , as inversion on the unit group need not be continuous with the subspace topology. (An example of this situation is theadele ring of a global field. Its unit group, called theidele group , is not a topological group in the subspace topology.) Embedding the unit group of "R" into the product "R" × "R" as ("x","x"^{-1}) does make the unit group a topological group. (If inversion on the unit group is continuous in the subspace topology of "R" then the topology on the unit group viewed in "R" or in "R" × "R" as above 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 which is a

topological group (for +) in which multiplication is continuous, too.**Examples**Topological rings occur in

mathematical analysis , for examples as rings of continuous real-valued functions on some topological space (where the topology is given by pointwise convergence), or as rings of continuouslinear operator s on somenormed vector space ; allBanach algebra s 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 number s anddual numbers form alternative topological rings. Seehypercomplex 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": asubset "U" of "R" is openif 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

integer s is an example of an "I"-adic topology (with "I" = ("p")).**Completion**Every topological ring is a

topological group (with respect to addition) and hence auniform space in a natural manner. One can thus ask whether a given topological ring "R" is complete. If it is not, then it can be "completed": one can find an essentially unique complete topological ring "S" which contains "R" as a densesubring such that the given topology on "R" equals the subspace topology arising from "S".The ring "S" can be constructed as a set of equivalence classes ofCauchy sequences in "R".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.**Topological fields**Some of the most important examples are also fields "F". To have a

**topological field**we should also specify that inversion is continuous, when restricted to "F"{0}. See the article onlocal field s for some examples.**References***springer|id=T/t093110|title=Topological ring|author=L. V. Kuzmin

*springer|id=T/t093060|title=Topological field|author=D. B. Shakhmatov

* Seth Warner: "Topological Rings". North-Holland, July 1993, ISBN 0444894462

* Vladimir I. Arnautov, Sergei T. Glavatsky and Aleksandr V. Michalev: "Introduction to the Theory of Topological Rings and Modules". Marcel Dekker Inc, February 1996, ISBN 0824793234.

* N. Bourbaki, "Éléments de Mathématique. Topologie Générale." Hermann, Paris 1971, ch. III §6

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