# Why are residue rings of prime ideals in algebraic number fields determined by their additive structure and the structure of their group of units?

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?

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I read through this and I'm a bit hung up on Thm 5 as well. In general, isomorphisms should be proven with actual maps. –  John M Aug 7 '11 at 13:31