Central traces of multiadditive maps on invertible matrices

Let be integers, and let be the ring of all matrices over a field with centre . Assume that is an m-additive map such that (), where denotes the trace map of G and is the set of all invertible matrices over . Then () if one of the followings holds: (1) char ; (2) char ; (3) char and ; (4) . As applications, we give a slight improvement in the theorem by Franca and extend the condition from the prime ring to its nonadditive subset for the theorems by Lee et al. and Beidar et al.