On Fillmore's theorem over integrally closed domains

2025/02/17 by Alexander Stasinski, Stasinski, Alexander
Mathematics · #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2502.11636

Abstract

A well-known theorem of Fillmore says that if A\inM_n(K) is a non-scalar matrix over a field K and γ₁,…,γ_n∈ K are such that γ₁+…+γ_n=Tr(A), then A is K-similar to a matrix with diagonal (γ₁,…,γ_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\inM_n(R) is non-scalar, then A is K-similar to a matrix in M_n(R) with diagonal (γ₁,…,γ_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\inM_n(R) to be R-similar to a matrix with diagonal (γ₁,…,γ_n). We show that when R is a PID and n≥3, Tan's condition is also sufficient.

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