Serre gives first the definition of a discrete valuation ring as a "principal ideal domain that has a unique non-zero prime ideal $m(A)$."
Next, he says: Let $A$ be a commutative ring. In order that A be a discrete valuation ring, it is necessary and sufficient that it be a Noetherian local ring, and that its maximal ideal be generated by a non-nilpotent element.
I do not understand why the unique non-zero prime ideal in a principal ideal domain is necessarily generated by a non-nilpotent element.
I started looking at the nilradical of a ring -- for a commutative ring, it is the intersection of all prime ideals. In this case we only have one prime ideal so the nilradical of our discrete valuation ring would be exactly the unique non-zero prime ideal given by the definition of a DVR. The nilradical is made up only by nilpotent elements, though, so surely it is generated by a nilpotent element, if we are in a principal ideal domain. To me, this would suggest that the maximal ideal of a DVR (by the first definition) is its nilradical, and it would be generated by a nilpotent element.
Clearly, I have made some mistake. Can somebody help me understand where I've gone wrong?