Additive preservers of rank-additivity on matrix spaces

Let F be a field. Let V denote the vector space of all m x n matrices over F or the vector space of all n x n symmetric matrices over F of characteristic not 2 or 3. For each fixed positive integer s >= 2, let Qs denote the set of all matrix pairs (A, B) in V such that rank(A + B) = rank(A) + rank(B) <= s. We characterize additive mappings psi on V such that (psi(A), psi (B)) epsilon Qs whenever (A, B) epsilon Qs for a fixed s. We also describe the structure of linear mappings from the space of n x n matrices over F to the space of p x q matrices over F that preserve rank-additivity, where char F not equal 2. (c) 2005 Elsevier Inc. All rights reserved.