Rank-one nonincreasing mappings on triangular matrices and some related preserver problems

Let T-n(F) be the algebra of all n x n upper triangular matrices over an ar bitrary field F. We first characterize those rank-one nonincreasing mapping s psi: T-n(F) --> T-m(F), n less than or equal to m such that psi (I-n) is of rank n. We next deduce from this result certain types of singular rank-o ne r-potent preservers and nonzero r-potent preservers on T-n(F). Character izations of certain classes of homomorphisms and semi-homomorphisms on T-n( F) are also given.