Valuation ring

A valuation ring is an integral domain $R$ such that, for every nonzero element $x$ of its fraction field $K$, either $x\in R$ or $x^{-1}\in R$. Equivalently, its ideals are totally ordered by inclusion.

A discrete valuation ring is a principal ideal domain with exactly one nonzero maximal ideal. Every nonzero element of its fraction field is a unit times a unique integer power of a uniformizer.

A discrete valuation on a field $K$ is a surjective group homomorphism $v:K^{\times }\to \mathbb{Z}$ satisfying $v(x+y)\ge \min (v(x),v(y))$ whenever $x+y\ne 0$. Its valuation ring is $\{0\}\cup \{x:v(x)\ge 0\}$.

The value group of a valued field is the ordered abelian group formed by the values of its nonzero elements. A valuation is discrete when its value group is infinite cyclic with the order inherited from $\mathbb{Z}$.

A uniformizer of a discrete valuation ring $R$ is a generator $\pi$ of its unique nonzero maximal ideal. Every nonzero $x$ in the fraction field has a unique expression $x=u{\pi }^n$ with $u$ a unit and $n\in \mathbb{Z}$.