Factorizations of Upper Triangular Matrices with Non-Commutative Entries
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The study of factorization theory concerns writing elements of various algebraic structures as products of other unfactorable elements. Factorization in commutative semigroups has been well studied, but far less work has been done in the non-commutative setting. We consider the factorization of triangular matrices with non-commutative entries under usual matrix multiplication. In particular, we provide a construction of a semigroup of block triangular matrices with diagonal entries from matrix semigroups of arbitrary sizes but over the same underlying ring. We then relate factorization properties in this semigroup to easier-to- understand semigroups, in particular products of semigroups, by means of a new tool, the weak transfer homomorphism.