Endomorphisms of upper triangular matrix rings

We investigate the class of endomorphisms alpha of a ring UT M-n(R) of upper triangular n x n matrices such that alpha(eij) is a (0,1)-matrix for any matrix unit e(ij). We use the left and right semicentral idempotents defined and studied by Birkenmeier. We study the idempotent semigroup (E-n(R), .) of endomorphisms of UT M-n(R). An endomorphism alpha is called regular if alpha(e(ii)) = e(ij )or alpha(e(ij)) = 0 for all i = 1, ... , n. In the main results we prove that the class of regular (0,1)-endomorphisms is E-n(R), that the semigroup (En(R), .) consists of all idempotent (0,1)-endomorphisms and all other (0,1)-endomorphisms are roots of idempotents.