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 for $R$ is a filed, 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.