Factorizations in self-idealizations of PIRs and UFRs

Abstract

The self-idealization of a commutative ring R is isomorphic to the ring \(R[x]/(x^2)\) or, equivalently, the ring of upper-triangular Toeplitz matrices \({{\mathrm{\mathcal {T}}}}(R)=\left\{ \left( \begin{matrix} a &{} b \\ 0 &{} a \end{matrix}\right) :a,b\in R\right\} \). Recently, Chang and Smertnig characterized the sets of lengths of factorizations in \({{\mathrm{\mathcal {T}}}}(D)\) where D is a principal ideal domain. In this work, in addition to correcting an error in their paper, we extend the study to \({{\mathrm{\mathcal {T}}}}(R)\) when R is either a principal ideal ring or a unique factorization ring.