It's known that, if $K$ is an infinite field, every linear mapping $T : \mathcal M_n(K) \rightarrow \mathcal M_n(K)$ which preserves determinant also preserves rank. I'm wondering if it is still true for a finite field.
Thank you for your help, Marcin
PS. We have $\det(T(M))=\det(M)$ for all $M\in \mathcal M_n(K)$ and I'm asking if $\mathrm{rg}(T(M))=\mathrm{rg}(M)$.