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Insights Into Principal Ideal Rings and Their Hereditary Properties

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  • Jin Xie
  • Kui Hu
  • Hwankoo Kim
  • DeChuan Zhou

Abstract

In this paper, we investigate principal ideal rings (PIRs). Specifically, we prove that every local PIR is either a 2-strongly Gorenstein semisimple ring or a discrete valuation ring, which leads to the establishment of the Gorenstein hereditary property for PIRs. In particular, we show that every PIR is G-hereditary. Furthermore, using pullbacks and techniques from generalized linear algebra, we provide an alternative proof of a classical result originally obtained by Krull. As a byproduct, we establish a new equivalent characterization of regular PIRs: a commutative ring R is a regular PIR if and only if every regular prime ideal of R is principal.

Suggested Citation

  • Jin Xie & Kui Hu & Hwankoo Kim & DeChuan Zhou, 2025. "Insights Into Principal Ideal Rings and Their Hereditary Properties," Journal of Mathematics, Hindawi, vol. 2025, pages 1-6, March.
    • Handle: RePEc:hin:jjmath:1217553
    DOI: 10.1155/jom/1217553
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