triangularization over finitedimensional division rings using the reduced trace

In this paper we study triangularization of collections of matrices whose entries come from a finite-dimensional division ring. first, we give a generalization of guralnick’s theorem to the case of finite-dimensional division rings and then we show that in this case the reduced trace function is a suitable alternative for trace function by presenting two triangularization results. the first one is a generalization of a result due to kaplansky and in the second one a triangularizability condition which is dependent on a single element is presented.