Factorization in rings of upper triangular Toeplitz matrices
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Let D be a principal ideal domain and let Rm(D) be the ring of (m + 1) × (m + 1) upper triangular Toeplitz matrices with entries in D. Then Rm(D) is a commutative, Noetherian ring with identity, and hence is atomic. We are able to classify many irreducible elements of Rm(D). Since Rm(D) is atomic, an element of Rm(D) can certainly be written as a product of irreducible elements. In this paper we determine how an element of Rm(D) might factor into a product of irreducible elements and are then able to determine bounds for the minimum and maximum lengths of factorizations of elements in Rm(D).