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Additive rank-one preservers between spaces of rectangular matrices

Suppose F is a field and m, n, p, q are positive integers. Let Mmn (F) be the set of all m × n matrices over F, and let Mmn 1(F) be its subset consisting of all rank-one matrices. A map ϕ : Mmn (F) → Mpq (F) is said to be an additive rank-one preserver if and ϕ(A + B) = ϕ(A) + ϕ(B) for any A, B ∈ Mmn (F). This article describes the structure of all additive rank-one preservers from Mmn (F) to Mpq (F).