# A two-liner

Is there a ring $A$ such that there are two different ring injections $\Bbb{R}\to A$?

1. $\Bbb{C}$ is such a ring. Take the non-trivial automorphism of $\Bbb{Q}[\sqrt{2}]/\Bbb{Q}$ and extend it to an automorphism $\Bbb{C}/\Bbb{Q}$. The restriction of the latter automorphism to $\Bbb{R}$ gives a non-trivial injection.
Notice that the image of $\Bbb{R}$ by this automorphism cannot be $\Bbb{R}$, since there are no non-trivial automorphisms of $\Bbb{R}$.
2. Easier solution, proposed by Iván: consider the maps $\Bbb{R}\to\Bbb{R}\otimes_{\Bbb{Z}} \Bbb{R}$ given by $x\mapsto x\otimes 1$ and $x\mapsto 1\otimes x$.