Matrices that are integral over the base ring

We study matrices over arbitrary rings that are integral over the base ring. For any ring $R$, let $f,g,h\in R[x_1,x_2,\dots ,x_n]$ with $h=fg$. We prove that for any pairwise commuting elements $a_1,a_2,\dots ,a_n\in R$ if $g(a_1,a_2,\dots ,a_n)=0$, then $h(a_1,a_2,\dots ,a_n)=0$. As a corollary, it follows that for $f\in {𝕄}_n(S)[x_1,x_2,\dots ,x_n]$, $S$ commutative ring, if $A_1,A_2,\dots ,A_n\in {𝕄}_n(S)$ are pairwise commuting matrices such that $f(A_1,A_2,\dots ,A_n)=0$, then $g(A_1,A_2,\dots ,A_n)=0$ where $g=\mathrm{\text{det}}(f)I$. This result, which is a generalization of the Cayley–Hamilton Theorem, was proved by Phillips in 1919. For a positive integer $n>1$, we prove that if every matrix in ${𝕋}_n(R)$ satisfies a monic polynomial of degree $n$ over $R$, then $R$ is commutative. On the other hand, every diagonal matrix in ${𝕄}_n(R)$, $n>1$, satisfies a monic polynomial of degree $n$ over $R$ precisely when $R$ is a left duo ring. We prove that if every diagonal matrix in ${𝕄}_n(R)$, $n>1$, is $R$-integral, then $R$ is Dedekind finite.