I didn't see this problem before. I motivated by the questions

Suppose A is an Abelian group such that it is possible to define an associative unitary ring structure on A. An element $x\in A$ is called potentially identity if there exists an associative ring $R$ such that $A$ is the additive group of $R$ and $x=1_R$, the identity element of $R$. Determine the set of all potentially identity elements of $A$.