Rank-permutable additive mappings

Let σ be a fixed non-identical permutation on k elements. Additive bijections T on the matrix algebra over a field of characteristic zero, with the property that rk(A1 Ak)=rk(Aσ(1) Aσ(k)) implies the same condition on the T images, are characterized. It is also shown that the surjectivity assumption can be relaxed, if this property is preserved in both directions.