Ring homomorphism from $M_3(\mathbb{R})$ into $\mathbb{R}$ I was working on this problem 

Find all ring homomorphisms from $M_3(\mathbb{R})$ into $\mathbb{R}$.

My attempt:-
I found that if we have any ring homomorphism $\phi$, then $\ker(\phi)$ should be either zero or the entire ring (since $M_3(\mathbb{R})$ is simple) and in case the ideal is the entire ring the ring homomorphism should be zero mapping. But I am  not sure if we have case where $\ker(\phi)={0}$. 

It is known that every injective ring homomorphism $\phi:M_n(K)\rightarrow M_m(K)$ must satisfy $n\le m$, see for example Lemma $3.2$ here. For $m=1$ and $n=3$ this is a contradiction. One could also use the Amitsur-Levitzki theorem.
