Given some Abelian group $(G, +)$, does there always exist a binary operation $\*$ such that $(G, +, \*)$ is a ring? That is, $\*$ is associative and distributive:

\begin{align\*}
&a \* (b \* c) = (a\*b) \* c \\
&a \* (b + c) = a \* b + a \* c \\
&(a + b) \* c = a \* c + b \* c \\
\end{align\*}

We also might have multiplicative identity $1 \in G$, with $a \* 1 = 1 \* a = a$ for any $a \in G$. Multiplication may or may not be commutative.

Depending on the definition, the answer could be no in the case of the group with one element: then $1 = 0$. But the trivial ring is not a very interesting case. For cyclic groups the statement is certainly true, since $(\mathbb{Z}\_n, +, \cdot)$ and $(\mathbb{Z}, +, \cdot)$ are both rings. What about in general? Is there some procedure to give arbitrary abelian groups ring structure?