On Two Problems Of Matrix Space Derivation Mapping

The matrix preserver problems not only has a great theoretical and practical signi?cance but also has a broad background in practical applications for many of the problems in system control, the ?elds of mathematical statistical and di?erentical equations. Therefore, in matrix theory, the study of preserver problems is one of the very active research areas. In recent years, we have gained a lot of results on rank-1preserver problems and rank-k preserver problems. And the study of the base ?elds is changing.Suppose D is a division ring. Let fij(i = 1, 2, · · ·, m; j = 1, 2, · · ·, n) be maps from D to itself. Denote by Mmn(D) the set of all m × n matrices over D. When m = n, we simply denote it by Mn(D). Denote by Tn(D) the set of all n × n upper triangular matrices over D. If a map f : Mmn(D) → Mmn(D) is de?ned by f :(aij) →(fij(aij)) for any(aij) ∈ Mmn(D), then we say that f is induced by {fij}. We say that a pair of matrices(A, B) is rank-additive if rank(A + B) = rank A + rank B.The map f is called a rank-additivity preserver if(f(A), f(B)) is rank-additive for any rank-additive(A, B). If A is invertible, then f(A) is invertible, and A is not invertible implies f(A) is not invertible, we say f is preserving invertible in both directions.Using the form of induced maps preserving rank-1 on Mmn(D), in this article we determine the form of induced maps preserving rank-additivity on Mmn(D) and the form of induced maps preserving invertibility in both directions on Mn(D). Also,we obtained a result of induced maps preserving rank-1 on Tn(D).