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?
(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 $2$.