Which finite simple groups are unit groups?

We prove that if $G$ is a finite simple group which is the unit group of a ring, then $G$ is isomorphic to either (a) a cyclic group of order 2; (b) a cyclic group of prime order $2^k-1$ for some $k$; or © a projective special linear group $PS{L}_n({\mathbb{F}}_2)$ for some $n\ge 3$. Moreover, these groups do (trivially) all occur as unit groups. We deduce this classification from a more general result, which holds for groups $G$ with no non-trivial normal 2-subgroup.