Willian Franca
IME-USP
Commuting Maps on Some Subsets that are not Closed Under Addition.
Abstract: Let n be a natural number greater or equal than 2. 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 in M_n(K) are of form px+m(x), where p belongs to Z, m :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 ax+m(x).