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 $\ker(\phi)={0}$. 

There is no injective ring homomorphism since every matrix of the form $AB-BA$ must be mapped into $0$. To conclude, it is well known that there exist some $AB-BA \neq 0$.
