Topological rings occur in mathematical analysis, for examples as rings of continuous realvalued 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 padic numbers are also topological rings (even topological fields[?]) with their standard topologies.
In algebra, the following construction is common: one starts with a commutative ring R containing an ideal I, and then considers the Iadic topology on R: a subset U of R is open iff 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 Iadic topology is Hausdorff if and only if the intersection of all powers of I is the zero ideal (0).
The padic topology on the integers is an example of an Iadic topolgy (with I = (p)).
Every topological ring is a topological group (with respect to addition) and hence a uniform 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 dense subring 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 of Cauchy sequences in R.
The rings of formal power series and the padic integers are most naturally defined as completions of certain topological rings carrying Iadic topologies.
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