These days, I am playing around Skolem Noether for matrix algebras, and a question arises.
If $F$ is a field , a by-product of Skolem Noether implies that any $F$-algebra automorphism $\rho$ of $M_n(F)$ preserves the characteric polynomial, that is $\chi_{\rho(M)}=\chi_M$ for all $M\in M_n(F)$.
Question 1. Can we prove that any $F$-algebra automorphism $\rho$ of $M_n(F)$ preserves the characteric polynomial without using the fact that $\rho$ is inner ? (I would like to use this fact to give a constructive proof of Skolem Noether)
Question 2. More generally, let $R$ be a commutative ring with $1$, and let $\rho$ be an $R$-algebra automorphism. Is is true that $\rho$ preserves the characteristic polynomial ??
It seems to be true for integral domains: if $R$ is a domain, let $K$ be its quotient field. Then $\rho\otimes Id_K$ is an automorphism of $M_n(K)$ and hence is inner (even if $\rho$ isn't), so the result follows easily, since $M$ may be viewed as a matrix with values in $K$.
If the answer for Question 2 is positive in full generality, I would be grateful to have an elementary argument (or at least, as much as possible).