Preserving commutativity

Every commutativity preserving linear map on the algebra of all n×n matrices over an algebraically closed field F with characteristic 0 is either a Jordan automorphism multiplied by a nonzero constant and perturbed by a scalar type operator, or its image is commutative. The assumption of preserving commutativity can be reformulated as preserving zero Lie products. So, this theorem is an extension of the well-known result on the structure of Lie homomorphisms of matrix algebras. We first prove the result for the special case in which F is the complex field and then apply the transfer principle in Model Theoretic Algebra to extend it to the general case.