| Let Mn(K)be the ring of all n x n matrices over a field K.In this paper,it is proved that a map f:Mn(K)→Mn(K)is additive if and only if f(A+B)= f(A)+f(B)for all rank-s matrices A,B ε Mn(K)when s is a fixed positive integer such that n/2 ≤ s<n,which has been proved to be true for the case s = n in a recent paper by Xu,Pei and Yi.We also provide an example showing that the conclusion will not be true if s<n/2.Then we extend it to the ring consisting of upper triangular matrices.Let Tn(K)be the ring of all n×n upper triangular matrices over a field K.We have that a map f:Tn(K)→ Tn(K)is additive if and only if f(A + B)= f(A)+ f(B)for all invertible matrices A,B E ∈ Tn(K)when |K|>2.We also give an example showing that the conclusion will not be true when |K| =2. |
