On Fillmore's theorem over integrally closed domains

Abstract

A well-known theorem of Fillmore says that if $A\in\operatorname{M}_{n}(K)$ is a non-scalar matrix over a field $K$ and $\gamma_{1},\dots,\gamma_{n}\in K$ are such that $\gamma_{1}+\dots+\gamma_{n}=\operatorname{Tr}(A)$, then $A$ is $K$-similar to a matrix with diagonal $(\gamma_{1},\dots,\gamma_{n})$. Building on work of Borobia, Tan extended this by proving that if $R$ is a unique factorisation domain with field of fractions $K$ and $A\in\operatorname{M}_{n}(R)$ is non-scalar, then $A$ is $K$-similar to a matrix in $\operatorname{M}_{n}(R)$ with diagonal $(\gamma_{1},\dots,\gamma_{n})$. We note that Tan's argument actually works when $R$ is any integrally closed domain and show that the result cannot be generalised further by giving an example of a matrix over a non-integrally closed domain for which the result fails. Moreover, Tan gave a necessary condition for $A\in\operatorname{M}_{n}(R)$ to be $R$-similar to a matrix with diagonal $(\gamma_{1},\dots,\gamma_{n})$. We show that when $R$ is a PID and $n\geq3$, Tan's condition is also sufficient.