I have been trying to plough through this paper by Elia, Interlando and Rosenbaum, and there is something that I don't quite understand. In proving a structure theorem for residue rings of unramified primes, they prove that a given residue ring of a prime ideal, $\zeta(\beta^a)$ is isomorphic additively to some ring, say $A$, and then that its group of units is isomorphic to $A^{(x)}$. This apparently suffices to prove the structure theorem.
Why is this?
Looking through the stackexchange archives this doesn't to always be true in general: Does a finite ring's additive structure and the structure of its group of units determine its ring structure? What's different here?
