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Factorizations in self-idealizations of PIRs and UFRs

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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.

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Authors and Affiliations

  1. Department of Mathematics, University of St. Thomas, St. Paul, MN, 55105, USA

    Michael Axtell

  2. School of Computer Science and Mathematics, University of Central Missouri, Warrensburg, MO, 64093, USA

    Nicholas R. Baeth

  3. Department of Mathematics, Millikin University, Decatur, IL, 62522, USA

    Joe Stickles

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  1. Michael Axtell
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  2. Nicholas R. Baeth
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  3. Joe Stickles
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Correspondence to Nicholas R. Baeth.

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Axtell, M., Baeth, N.R. & Stickles, J. Factorizations in self-idealizations of PIRs and UFRs. Boll Unione Mat Ital 10, 649–670 (2017). https://doi.org/10.1007/s40574-016-0107-8

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  • DOI: https://doi.org/10.1007/s40574-016-0107-8

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