Ring with additive group isomorphic to group of units

Question

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?

Answer 1

(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 $2$.