TITLE: Are there any nontrivial ring homomorphisms $M_{n+1}(R)\rightarrow M_n(R)$?
QUESTION [27 upvotes]: 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.

REPLY [5 votes]: Since it is not clear that $R$ must be commutative (none of the tags eliminate this possibility), the finite generation hypothesis (which I interpret as finitely generated as an algebra over a field) is not  significant, in view of the existence of rings for which all finitely generated projective modules are cyclic and free. 
The standard dense subalgebra of the Cuntz C*-algebra is such an example, and is  finitely generated as an algebra over the complexes, and satisfies $M_n R $ isomorphic to $R$ for all $n$, yielding unital maps (isomorphisms) $M_{n+1}R \to M_n R$.