Factorizations of block triangular matrices

Transfer homomorphisms have long been used as tools in factorization theory. The idea is to transfer information about factorizations in a semigroup of interest from simpler, easier-to-understand semigroups. Unfortunately there exist noncommutative semigroups which admit no transfer homomorphism to any commutative semigroup. The weak transfer homomorphism was introduced to study such semigroups. In particular, these weak transfer homomorphisms were used to study factorizations of semigroups of upper-triangular matrices with entries coming from a commutative integral domain. In this work we extend this study, finding weak transfer homomorphisms to both commutative and noncommutative semigroups from upper triangular matrices with entries coming from noncommutative rings. In particular, we study factorizations of block triangular matrices.