Editing Discrete valuation ring (section)

==== Localization of Dedekind rings ====
Let <math>\mathbb{Z}_{(2)} := \{ z/n\mid z,n\in\mathbb{Z},\,\, n\text{ is odd}\}</math>.  Then, the field of fractions of <math>\mathbb{Z}_{(2)}</math> is <math>\mathbb{Q}</math>.  For any nonzero element <math>r</math> of <math>\mathbb{Q}</math>, we can apply [[fundamental theorem of arithmetic|unique factorization]] to the numerator and denominator of ''r'' to write ''r'' as {{sfrac|2<sup>''k''</sup> ''z''|''n''}} where ''z'', ''n'', and ''k'' are integers with ''z'' and ''n'' odd.  In this case, we define ν(''r'')=''k''.
Then <math>\mathbb{Z}_{(2)}</math> is the discrete valuation ring corresponding to ν. The maximal ideal of <math>\mathbb{Z}_{(2)}</math> is the principal ideal generated by 2, i.e. <math>2\mathbb{Z}_{(2)}</math>, and the "unique" irreducible element (up to units) is 2 (this is also known as a uniformizing parameter). Note that <math>\mathbb{Z}_{(2)}</math> is the [[localization of a ring|localization]] of the [[Dedekind domain]] <math>\mathbb{Z}</math> at the [[prime ideal]] generated by 2.

More generally, any [[Localization (commutative algebra)|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 [[ring (mathematics)|rings]]
:<math>\mathbb Z_{(p)}:=\left.\left\{\frac zn\,\right| z,n\in\mathbb Z,p\nmid n\right\}</math>
for any [[prime number|prime]] ''p'' in complete analogy.