Insights Into Principal Ideal Rings and Their Hereditary Properties

Jin Xie, Kui Hu, Hwankoo Kim and DeChuan Zhou

Journal of Mathematics, 2025, vol. 2025, 1-6

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.

Date: 2025
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jjmath:1217553

DOI: 10.1155/jom/1217553

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