Topology
Every discrete valuation ring, being a local ring, carries a natural topology and is a topological ring. The distance between two elements x and y can be measured as follows:
(or with any other fixed real number > 1 in place of 2). Intuitively: an element z is "small" and "close to 0" iff its valuation ν(z) is large.
A DVR is compact if and only if it is complete and its residue field R/M is a finite field.
Examples of complete DVRs include the ring of p-adic integers and the ring of formal power series over any field. For a given DVR, one often passes to its completion, a complete DVR containing the given ring that is often easier to study. This completion procedure can be thought of in a geometrical way as passing from rational functions to power series.
Returning to our examples: the ring of all formal power series in one variable with real coefficients is the completion of the ring of rational functions defined in a neighborhood of 0 on the real line; it is also the completion of the ring of all real power series that converge near 0. The completion of Z(p) (which can be seen as the set of all rational numbers that are p-adic integers) is the ring of all p-adic integers Zp.
Read more about this topic: Discrete Valuation Ring