Discrete valuation ring

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 2^kz ⁄ 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 ${\displaystyle \mathbb {Z} _{(p)}:=\left.\left\{{\frac {z}{n}}\,\right|z,n\in \mathbb {Z} ,p\nmid n\right\}}$ for any prime p in complete analogy.

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