TITLE: Generality of rings' abelian group
QUESTION [5 upvotes]: Let G be an abelian (finite) group. Is there a ring $R$ with $G$ isomorphic to the group $(R,+)$?

REPLY [5 votes]: 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.