Additive preservers of rank-additivity on matrix spaces

Let F be a field. Let V denote the vector space of all m × n matrices over F or the vector space of all n × 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 ψ on V such that (ψ(A), ψ(B)) ∈ Qs whenever (A, B) ∈ Qs for a fixed s. We also describe the structure of linear mappings from the space of n × n matrices over F to the space of p × q matrices over F that preserve rank-additivity, where charF≠2.