Let $R$ be a finitely generated ring with identity, $M\_n(R)$ the set of $n\times n$ matrices. Are there any nontrivial ring homomorphisms $M\_{n+1}(R)\rightarrow M\_n(R)$? This should be an elementary question in abstract algebra. But even if $R$ is a field, I couldn't get a quick (negative) proof. Any comments are welcomed.


RMK: If we view the natrual map $M\_{n}(R)\rightarrow M\_{n+1}(R)$ as a ring homomorphism, we will not require that a ring homomorphism preserves identities.



Are there any nontrivial ring homomorphisms $M_{n+1}(R)\rightarrow M_n(R)$?
According to the Amitsur-Levitzki theorem, $n \times n$ matrices over a commutative ring satisfy a polynomial identity of degree $2n$ and none of smaller degree. So there can be no injective ring homomorphism $M\_{n+1}(R) \to M\_n(R)$, which at least rules out the case when $R$ is a field.


