Frobenius extensions about centralizer matrix algebras

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 $\mathbb{k}$, 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 $\mathbb{k}$. We extend this analysis to arbitrary matrices over a field and derive conditions for matrix diagonalizability through Frobenius extensions.