Preserving commutativity

Every commutativity preserving linear map on the algebra of all n x n matri ces over an algebraically closed field F with characteristic 0 is either a Jordan automorphism multiplied by a nonzero constant and perturbed by a sca lar type operator, or its image is commutative. The assumption of preservin g commutativity can be reformulated as preserving zero Lie products. So, th is 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 specia l case in which F is the complex field and then apply the transfer principl e in Model Theoretic Algebra to extend it to the general case. (C) 2001 Els evier Science B.V. All rights reserved. MSC. 15A04; 15A27.