Let $R$ be a commutative ring with identity and $M_n(R)$ the set of $n$ by $n$ matrices over $R$. Let $C_A(X)$ be the characteristic polynomial of $A$. Denote by $N_A$ the set (ideal) $N_A=\{p(X)\in R[X]\;\;:\;\;p(A)=0\}.$
Show that $(C^\prime_A(X))+N_A=R[X]$ implies that $R[A]=${$p(A)\;:\;p(X)\in R[X]$} are the only matrices in $M_n(R)$ which commute with $A$.
Note: $(C^\prime_A(X))$ is the ideal generated by $C^\prime_A(X)$, the formal derivative of $C_A(X)$
This is an exercise of: William Brown "Matrices over Commutative Rings"