Centralizers and their applications to generalized inverses

摘要

Let M_n (D) be the ring of all n × n matrices over a division ring ID, where n ≥ 2 is an integer and let GL_n（D） be the set of all invertible matrices in M_n (D). We describe maps f: GL_n (D) → M_n (D) such that [f(x),f(y）] = [x,y] for all x, y ∈ GL_n (D). The analogous result for singular matrices is also obtained.