If $A \in M_n(R)$, with $R$ a P.I.D., can $A$ be put in Jordan form iff all the roots of the characteristic polynomial are in $R$?

Question

If $A \in M_n(R)$, with $R$ a P.I.D., can $A$ be put in Jordan form iff all the roots of the characteristic polynomial are in $R$?

If this is false in general, is it possibly true for nilpotent matrices?

Answer 1

Let $R$ be the set of integers. If the integer $k$ is a factor of $A$ then it is a factor of $P^{-1}AP$.