and every other non-zero ideal is a power of M, i.e. has the form (t k) for some k≥0. All the powers of t are distinct, and so are the powers of M. Every non-zero element x of R can be written in the form αt k with α a unit in R and k≥0, both uniquely determined by x. The valuation is given by ν(x) = kv(t). So to understand the ring completely, one needs to know the group of units of R and how the units interact additively with the powers of t. The function v also makes any discrete valuation ring into a Euclidean domain. Topology Every discrete valuation ring, being a local ring, carries a natural topology and is a topological ring. We can also give it a metric space structure where 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. The function |x-y|, supplemented by |0|=0, is the restriction of an absolute value defined on the field of fractions of the discrete valuation ring. 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, or from rational numbers to the reals. 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 (i.e. finite) 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 (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. See also :Category:Localization (mathematics) Local ring Ramification of local fields Cohen ring Valuation ring References Discrete valuation ring, The Encyclopaedia of Mathematics. Commutative algebra Localization (mathematics)