Frobenius extensions about centralizer matrix algebras

Qikai Wang; Haiyan Zhu

doi:10.1016/j.laa.2025.07.002

paper

Frobenius extensions about centralizer matrix algebras

arXiv:2602.17328 · doi:10.1016/j.laa.2025.07.002

Abstract

This paper investigates the conditions under which the centralizer algebra $S_{n} (c, R)$ of a matrix $c \in M_{n} (R)$ is a (separable) Frobenius extension of the base algebra $R$ . For an algebra $R$ over an integral domain $k$ , we provide necessary and sufficient conditions for $S_{n} (c, R) / R$ to be a (separable) Frobenius extension when $c$ is in Jordan canonical form with eigenvalues in $k$ . We extend this analysis to arbitrary matrices over a field and derive conditions for matrix diagonalizability through Frobenius extensions.