For which positive integer $n$ does there exist a $\Bbb R$-linear ring homomorphism $f:\Bbb C \to M_n(\Bbb R)?$
If I send everything to zero matrix that will give trivial ring homomorphism for every $n$ .
Now nontrivial ring homomorphism: for $n=1$ there is no such homomorphism from $\Bbb C$ to $M_n(\Bbb R)$ as there is no ring homomorphism from $\Bbb C$ to $\Bbb R$. Now $M_n(\Bbb R)$ is not an integral domain for $n\ge2$, so $f(1)$ will be an idempotent element. I am not getting any idea to proceed further.
Thanks in advance.