I will be so thankful if someone can help me with the following question.

Is it possible to obtain all maximal centralizers in the full matrix ring, $M_n(F)$, for an arbitrary finite field $F$? Here, by maximal centralizer I mean: $C=C_R(x)$ is maximal if $C\subseteq C_R(y)$, then $C=C_R(y)$ or $y\in Z(R)$.

In Akbari et al., Linear Alg. App. 390 (2004) 345-355, lemma 3 determines all centralizers with maximum dimension. So some of the maximal centralizers are determined. Is it true that the set of centralizers with maximum dimension and the set of maximal centralizers are equal?

Hamid.