  Article Content

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 + InU. 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.

All Wikipedia text is available under the terms of the GNU Free Documentation License

Search Encyclopedia
 Search over one million articles, find something about almost anything!

Featured Article
 List of intelligence agencies ... and Security[?] Iraq General Intelligence Directorate[?] Special Security Service[?] Israel ha-Mossad le-Modiin ule-Tafkidim Meyuhadim (Mossad) Sherut ...  