Finite Rings whose additive structure is isomorphic to $\mathbb{Z}/(n \mathbb{Z})$ I am having trouble proving the following conjecture: If $R$ is a ring with $1_R$ different from $0_R$ s.t. its additive structure is isomorphic to  $\mathbb{Z}/(n \mathbb{Z})$ for some $n$, must $R$ always be isomorphic to the ring  $\mathbb{Z}/(n \mathbb{Z})$ ? How do we go about defining a ring isomorphism with a proper multiplication on $R$? 

Combine the following general facts:
For any ring $R$, the prime ring (i.e. the subring generated by $1$) is isomorphic to the quotient of $\mathbb Z$ by the annihilator of $R$ in $\mathbb Z$. 
Any cyclic group $R$ is isomorphic to the quotient of $\mathbb Z$ by the annihilator of $R$ in $\mathbb Z$.
(This is Mariano's answer with slightly different words.)
