Discrete valuation ring

In abstract algebra, a discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non-zero maximal ideal. In abstract algebra, a discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non-zero maximal ideal. This means a DVR is an integral domain R which satisfies any one of the following equivalent conditions: Let Z(2) := { z / n : z, n ∈ Z, n odd }. Then the field of fractions of Z(2) is Q. Now, for any nonzero element r of Q, we can apply unique factorization to the numerator and denominator of r to write r as 2k z/n where z, n, and k are integers with z and n odd. In this case, we define ν(r)=k.Then Z(2) is the discrete valuation ring corresponding to ν. The maximal ideal of Z(2) is the principal ideal generated by 2, and the 'unique' irreducible element (up to units) is 2. Note that Z(2) is the localization of the Dedekind domain Z at the prime ideal generated by 2. Any localization of a Dedekind domain at a non-zero prime ideal is a discrete valuation ring; in practice, this is frequently how discrete valuation rings arise. In particular, we can define rings Z ( p ) := { z n | z , n ∈ Z , p ∤ n } {displaystyle mathbb {Z} _{(p)}:=left.left{{frac {z}{n}}, ight|z,nin mathbb {Z} ,p mid n ight}} for any prime p in complete analogy. For an example more geometrical in nature, take the ring R = {f/g : f, g polynomials in R and g(0) ≠ 0}, considered as a subring of the field of rational functions R(X) in the variable X. R can be identified with the ring of all real-valued rational functions defined (i.e. finite) in a neighborhood of 0 on the real axis (with the neighborhood depending on the function). It is a discrete valuation ring; the 'unique' irreducible element is X and the valuation assigns to each function f the order (possibly 0) of the zero of f at 0. This example provides the template for studying general algebraic curves near non-singular points, the algebraic curve in this case being the real line. Another important example of a DVR is the ring of formal power series R = k [ [ T ] ] {displaystyle R=k]} in one variable T {displaystyle T} over some field k {displaystyle k} . The 'unique' irreducible element is T {displaystyle T} , the maximal ideal of R {displaystyle R} is the principal ideal generated by T {displaystyle T} , and the valuation ν {displaystyle u } assigns to each power series the index (i.e. degree) of the first non-zero coefficient. If we restrict ourselves to real or complex coefficients, we can consider the ring of power series in one variable that converge in a neighborhood of 0 (with the neighborhood depending on the power series). This is also a discrete valuation ring. Finally, the ring Z p {displaystyle mathbb {Z} _{p}} of p-adic integers is a DVR, for any prime p {displaystyle p} . Here p {displaystyle p} is an irreducible element; the valuation assigns to each p {displaystyle p} -adic integer x {displaystyle x} the largest integer k {displaystyle k} such that p k {displaystyle p^{k}} divides x {displaystyle x} . For a DVR R {displaystyle R} it is common to write the fraction field as K = Frac ( R ) {displaystyle K={ ext{Frac}}(R)} and κ = R / m {displaystyle kappa =R/{mathfrak {m}}} the residue field. These correspond to the generic and closed points of Spec ( R ) {displaystyle { ext{Spec}}(R)} . For example, the closed point of Spec ( Z p ) {displaystyle { ext{Spec}}(mathbb {Z} _{p})} is F p {displaystyle mathbb {F} _{p}} and the generic point is Q p {displaystyle mathbb {Q} _{p}} . Sometimes this is denoted as