I'm doing problem 2.26 in the book "algebraic curves" by Fulton:
One is given two DVR's R and S both of which have the same field of fractions $K$. It is assumed that the maximal ideal of $R$ is contained in that of $S$, and the problem is to show that $R$ and $S$ are equal.
I managed to show that if an element of $S$ is in the maximal ideal of $S$ but not in the maximal ideal of $R$, then its inverse is also in the maximal ideal of $S$, which implies that $S=K$, a contradiction (as a DVR may not be a field).
My question is then, having showed that $R, S$ must have the same maximal ideal, how does it follow that $R=S$, i.e., what about the units?