Factorizations of upper triangular Toeplitz matrices

Abstract

Let D be a principal ideal domain and m a positive integer. We denote by \(R_m(D)\) the ring of \((m+1) \times (m + 1)\) upper triangular Toeplitz matrices with entries in D. Then \(R_m(D)\cong R[X]/(X^{m+1})\) is a commutative Noetherian ring with identity and thus every nonzero non-unit of \(R_m(D)\) can be written as a finite product of irreducible elements. In this manuscript we give conditions for an element of \(R_m(D)\) to be irreducible, describe the general structure of factorizations of elements as products of irreducible elements, and give a description of how certain elements factor as products of irreducible elements. In particular, we generalize work of Chang and Smertnig in the case where \(m=1\).