Discrete valuation ring

For a local domain $(R,\mathfrak m)$ that is not a field, the following are equivalent:

1. $R$ is a DVR.
2. $R$ is a principal ideal domain and has a unique nonzero prime ideal (namely $\mathfrak m$).
3. There exists $\pi\in \mathfrak m$ such that every nonzero ideal of $R$ is of the form $(\pi^n)$ for a unique $n\ge 0$.

In particular, in a DVR the ideals form a totally ordered chain:

DVRs arise naturally from Dedekind domains: if $R$ is Dedekind and $\mathfrak p\neq (0)$ is prime, then $R_\mathfrak p$ is a DVR (via localization at $\mathfrak p$).

1. Formal power series. If $k$ is a field, then $k[[t]]$ is a DVR with maximal ideal $(t)$ and uniformizer $t$.

1. Localization of a PID at a prime. $\mathbb{Z}_{(p)}$ is a DVR: it is local with maximal ideal generated by $p$, and every ideal is $(p^n)$.

1. A polynomial example. $k[t]_{(t)}$ (localization of $k[t]$ at the prime $(t)$) is a DVR with uniformizer $t$.

In a DVR, the valuation “order of vanishing” is encoded by powers of $\pi$: for $0\ne x\in \mathrm{Frac}(R)$, one can write $x=\pi^n u$ with $u\in R^\times$ and $n\in\mathbb{Z}$, uniquely.