On Gerstenhaber’s theorem for spaces of nilpotent matrices over a skew field

Let K be a skew field, and K0 be a subfield of the central subfield of K such that K has finite dimension q over K0. Let V be a K0-linear subspace of n×n nilpotent matrices with entries in K. We show that the dimension of V is bounded above by , and that equality occurs if and only if V is similar to the space of all n×n strictly upper-triangular matrices over K. This generalizes famous theorems of Gerstenhaber and Serezhkin, which cover the special case K=K0.