Let G be an abelian (finite) group. Is there a ring $R$ with $G$ isomorphic to the group $(R,+)$?
Even if you require rings to have $1$, since every finite abelian group is isomorphic to the direct product of $\mathbb Z/n\mathbb Z$'s, you can just extend this into a ring in the obvious way, where the $1$ is achieved by letting each factor equal $1$.
The result is still true even if you ask that $G$ is finitely generated. This is because every finitely generated abelian group is product of a finite number of cyclic groups which in turn are isomorphic other to $\Bbb Z$ or $\Bbb Z/n\Bbb Z$, and these are also rings.