Factorizations of upper triangular matrices

If D   is an integral domain in which every element can be factored as a product of irreducible elements, then every element A∈Tn(D)A∈Tn(D), the semigroup of upper triangular matrices with nonzero determinant, can be factored as a product of irreducible elements. We classify both the irreducible elements and units of this cancellative semigroup. Having achieved this, we provide a means of measuring the uniqueness or non-uniqueness of these factorizations in terms of the uniqueness or non-uniqueness of factorizations in D. In addition to using well-studied invariants from factorization theory of commutative semigroups, we introduce a new tool, weak transfer homomorphisms, which are more appropriate for noncommutative settings.