I will be so thankful if some one help me about the following question.

Is it possible to obtain all maximal centralizers in full matrix ring, $M_n(F)$, for arbitrary finite field $F$, where 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, in Lemma 3 determines all centralizers with maximum dimension. So some of maximal centralizers are determined. Is it true that the set of centralizer with maximum dimension and the set of maximal centralizers are equal? Hamid.