Similarity And Commutators Of Gauss Integral Matrices

Let F be a field,Mn(F)the algebra of n × n matrices over F and A ? Mn(F)with trace(A)=0.The following facts are well-known(i)if A is not a scalar,then A is similar over F to a matrix with zero diagonal(ii)A=[P,Q]=PQ-QP for some P,Q ? Mn(F).In this paper we consider the situation when F is replaced by the ring of Gauss integers Z[i].We find that(ii)holds,(i)holds if F for n>2 provided A(?)al(mod p)for all integers a and primes p.