discrete valuation ring
A discrete valuation ring is the algebra of 'order of vanishing' at a single smooth point on a curve. Near such a point every nonzero function vanishes to some whole-number order — once, twice, or to order zero if it does not vanish — and a single function, the uniformizer, vanishes to order exactly one. A DVR packages this: it is the cleanest possible one-dimensional local ring, where divisibility is measured by one integer and every ideal is a power of the maximal one.
A discrete valuation on a field K is a surjective map v: K^* -> Z satisfying v(xy) = v(x) + v(y) and v(x + y) >= min(v(x), v(y)). Its valuation ring R = { x : v(x) >= 0 } ∪ {0} is the associated DVR. Equivalently, a DVR is a local principal ideal domain that is not a field; equivalently a Noetherian local domain of dimension one that is integrally closed; equivalently a one-dimensional regular local ring. The maximal ideal is (t) for a uniformizer t, and every nonzero ideal is (t^n).
DVRs are the local building blocks of Dedekind domains: localizing a Dedekind domain at a nonzero prime always yields a DVR, and this is how unique factorization of ideals becomes, locally, just 'count the power of t'. The honest contrast: a one-dimensional Noetherian local domain that is NOT integrally closed (such as the node k[x, y]/(y^2 - x^3) localized at the origin) fails to be a DVR — smoothness, i.e. regularity, is exactly the missing ingredient.
The ring k〚t〛 of formal power series over a field k is a DVR: v(f) = order of the lowest nonzero term. Here t is the uniformizer, the maximal ideal is (t), and the nonzero ideals are (t), (t^2), (t^3), ....
Power series: valuation = order of vanishing.
The p-adic integers Z_p are a DVR with uniformizer p and valuation v_p (the p-adic valuation); their fraction field is the p-adic numbers Q_p. Every DVR is a local UFD with exactly one prime element up to units.