Suppose $R$ is a noncommutative ring. When could I reasonably expect $R^{op}\cong R$?
For instance I know group rings have a natural involution, i.e. if $SG$ is the group ring in question then $r\cdot g=g^{-1}\cdot r$. Does this imply $S[G]^{op}\cong S[G]$? If so, does this apply to any ring with a natural involution?
Finally, does $R^{op}\cong R$ imply all left ideals are right ideals (and vice versa)?