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 Mmn1(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 φ(Mmn(F)) ⊆ Mpq1(F) and φ(A + B) = φ(A) + φ(5) for any A, B ∈ Mmn(F). This article describes the structure of all additive rank-one preservers from Mmn(F) to Mpq(F).