Semicentral idempotents of upper triangular matrix rings

In this paper, we describe the necessary and sufficient conditions for upper triangular [Formula: see text] matrix over a ring to be left (right) semicentral idempotent. Circle compositions of left and right semicentral idempotent matrices ([Formula: see text]) are considered. Many researchers deal with the problems of expressing various matrices as a product, a sum or a linear combination of some idempotents. Following the results of A. Alahmadi, S. K. Jain, A. Leroy and A. Sathaye for decompositions into products of idempotents, we represent an arbitrary strictly upper triangular matrix as a product of circle compositions. These circle compositions are sums since the considered left and right semicentral idempotents are orthogonal matrices.