Rank-one non-increasing additive maps between block triangular matrix algebras

Let m and n be positive integers with n m. In this note, we characterize rank-one non-increasing additive maps from n-square block triangular matrix algebras to m-square matrix algebras over an arbitrary field with the assumption that (In) is of rank n. We deduce from this result a complete classification of adjugate-commuting additive maps between block triangular matrix algebras. As a corollary, we characterize adjugate-commuting additive maps between square matrix algebras.