For a positive integer $n$, I have to show that there does not exist a onto ring homomorphism from $M_{n+1 \times n+1}(\mathbb F) \to M_{n \times n}(\mathbb F) $ for any field $\mathbb F.$
If it exists lets say there is $f:M_{n+1 \times n+1}(\mathbb F) \to M_{n \times n}(\mathbb F)$ surjection and since $1$ maps to $1,$ its kernel is trivial and hence an isomorphism. Now if I can show that $f$ is $\mathbb F$- linear then we are done by dimension argument. To show that $f$ is $\mathbb F$-linear enough to show that $f(cI_{n+1})=cI_n$ for each $c \in \mathbb F,$ which I am not able to show. Or may be there is some alternative way to prove it. I need some help to show it.