Commuting maps on rank-k matrices

Let n >= 2 be a natural number. Let M-n(K) be the ring of all n x n matrices over a field K. Fix natural number k satisfying 1 < k <= n. Under a mild technical assumption over K we will show that additive maps G : M-n (K) -> M-n(K) such that [G(x), x] = 0 for every rank-k matrix x is an element of M-n (K) are of form lambda x + mu(x), where lambda is an element of Z, mu : M-n(K) -> Z, and Z stands for the center of M-n (K). Furthermore, we shall see an example that there are additive maps such that [G(x), x] = 0 for all rank-1 matrices that are not of the form lambda x + mu(x). (C) 2012 Elsevier Inc. All rights reserved.