Generalized Commuting Maps On The Set of Singular Matrices

Abstract

 Let Mn(K) be the ring of all n × n matrices over  a field K. In the present paper, additive mappings G : Mn(K) → Mn(K) such that [[G(y), y], y] = 0 for all singular matrix y will be characterized. Precisely, it will be proved that G(x) = λx + µ(x) for all x ∈ Mn(K), where λ ∈ K and µ is a central map.  As an application, the description is given of all  additive  maps  g : Mn(K) → Mn(K)  such that [[g(y^k1 ), y^k2 ], y^k3 ] = 0 for all singular matrices y ∈ Mn(K),  where m ∈ N^∗.