Earlier I went over the definition and first properties of a discrete valuation ring.  Today, it’s time to say how we can tell a ring is a DVR–it turns out to be not too bad, which is nice because the properties we need in this criterion are often easier to work with than the existence of some discrete valuation.

Today’s result is:

Theorem 1 If the domain ${R}$ is Noetherian, integrally closed, and has a unique nonzero prime ideal ${\mathfrak{m}}$, then ${R}$ is a DVR. Conversely, any DVR has those properties. (more…)