Insights Into Principal Ideal Rings and Their Hereditary Properties

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.