Is there a Ring $R$ with $(R,+) \cong (R^\times,\cdot)$? If $R$ is finite, clearly only the trivial ring does it (for cardinality reasons). But what about infinite rings? Are there even fields as example?
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(as pointed out in the comments, this proof works only when $R$ is a field)
By the given isomorphism, the equations $2x=0$ and $x^2=1$ have the same number of solutions. But $2x=0$ has nontrivial solutions if and only if $R$ has characteristic $2$, while $x^2=1$ has nontrivial solutions if and only if $R$ has characteristic different from $2$.
Andrew Dudzik
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The first paragraph doesn't hold - the initial question asked for an isomorphism between $(R, +)$ and the group of units of $R$, not the monoid of non-zero elements of $R$. But this does answer the secondary question of the case where $R$ is a field. – Christopher Jul 18 '18 at 09:10