From the theorem of classification of finitely generated abelian groups, we can see that every finitely generated commutative group can be considered as the additive structure underlying (at least) one unitary and commutative ring. My question is about possible generalization of
this result.
Question: Is it true that with every abelian group G it is possible to build (at least) one ring(resp: unitary ring, commutative ring) whose additive structure is the group structure
 on G ?
Gérard Lang



Is every commutative group structure underlying at least one (unitary, commutative) ring structure
As Sasha Anan'in already mentioned, there are counterexamples like $\mathbb{Z}/p^\infty = \mathbb{Z}[\frac{1}{p}]/\mathbb{Z}$ and $\mathbb{Q}/\mathbb{Z}$. The common feature of these groups is that there are divisible *and* torsion and therefore $A\otimes\_\mathbb{Z}A=0$. A ring $A$ has a surjective and hence non-zero map $A\otimes\_\mathbb{Z}A\to A$ namely the multiplication. Therefore no additive group with $A\otimes\_\mathbb{Z}A=0$ can be the additive group of a ring.


