A discrete valuation ring $R$ is a principal ideal domain with exactly one nonzero maximal ideal $M$ . Any generator $t$ of $M$ is called a uniformizer or uniformizing element of $R$ ; in other words, a uniformizer of $R$ is an element $t \in R$ such that $t \in M$ but $t \notin M^2$ .

Given a discrete valuation ring $R$ and a uniformizer $t \in R$ , every element $z \in R$ can be written uniquely in the form $u \cdot t^n$ for some unit $u \in R$ and some nonnegative integer $n \in \mathbb{Z}$ . The integer $n$ is called the order of $z$ , and its value is independent of the choice of uniformizing element $t \in R$ .