Is there an example of two groups $G\_{1}, G\_{2}$ such that there are
two non isomorphic ring $R\_{1}$ and $R\_{2}$ such that the additive group of both rings is isomorphic to $G\_{1}$ and their unit groups is isomorphic to $G\_{2}$?


Lets generalize this question as follows:


Is there an example of two groups $G\_{1}$ and $G\_{2}$ such that there are infinite number of non isomorphic rings which their additive and unit groups are isomorphic to $G\_{1}$ and $G\_{2}$, respectively?


this question is motivated by the following post:


[A basic question about rings](https://mathoverflow.net/questions/154852/a-basic-question-about-rings#comment396779_154852)



A question in ring theory
Take $\mathbb{R}$ and $\mathbb{R}[x]$. All units in the latter are constant polynomials, so the unit groups are isomorphic. The additive groups of both have continuum dimension as rational vector spaces, so they are isomorphic. But clearly they are not isomorphic. 


It seems the same argument shows that we could take polynomial rings over $\mathbb{R}$ in finitely many variables, and that gives an infinite class of nonisomorphic rings. 


