The role of semicentral idempotents in triangular matrix rings

The circle composition e1 degrees e2=e1+e2-e1e2 is well-known from the seminal book of Jacobson. A part of our motivation aims to find non-orthogonal idempotents e1 and e2 such that e1 degrees e2 is an idempotent. Such idempotents are the products rl where r is a right semicentral and l is a left semicentral idempotent of the ring of upper triangular matrices over a ring. We prove that in the semigroup of upper triangular matrices over ring with only trivial idempotents every idempotent matrix can be represented as a circle composition of products of the type rl.