Topological ring

In mathematics, a topological ring is a ring R which is also a topological space such that both the addition and the multiplication are continuous as maps R × R -> R (here R × R carries the product topology).

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 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) 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 I-adic 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 + Iⁿ ⊆ 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 topolgy (with I = (p)).

Completion

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 p-adic integers are most naturally defined as completions of certain topological rings carrying I-adic topologies.