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

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Let m and n be positive integers with n ≤ m. In this note, we characterize rank-one non-increasing additive maps ψ from n-square block triangular matrix algebras to m-square matrix algebras over an arbitrary field with the assumption that ψ(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.