In this post I will prove a commutative algebra lemma, which is proved in page 7 of Serre’s Local Fields. This lemma is useful if you find a ring in the wild and want to know that it’s a discrete valuation ring.
Proposition: Let be a noetherian local domain where and is not nilpotent. Then .
Proof: Suppose . Then for all . So for all . Since is a domain, .
Therefore consider the ascending chain . This eventually stabilizes for high enough since is noetherian, so for some , . Thus , so . But is a unit, so , so .
This theorem holds more generally even if is not assumed to be a domain, but the proof is more complicated (but still among the same lines).
Proposition: Let be a noetherian local ring where and is not nilpotent. Then .
Proof: Let be the ideal of elements that kill some power of . We will use variables to refer to elements of . Since is noetherian, must be finitely generated, so all elements of kill for some fixed .
Now suppose . , so . Thus .
Consider the ascending chain . Since is noetherian it must eventually stablize, so for some , can be written as . But recall that . So so . is a unit since , and is local, so . If we force to be large enough to surpass , then , so .