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Bull. Korean Math. Soc. 2024; 61(3): 735-744

Online first article May 20, 2024      Printed May 31, 2024

https://doi.org/10.4134/BKMS.b230323

Copyright © The Korean Mathematical Society.

A Gorenstein homological characterization of Krull domains

Shiqi Xing, Xiaolei Zhang

Chengdu University of Information Technology; Shandong University of Technology

Abstract

In this note, we shed new light on Krull domains from the point view of Gorenstein homological algebra. By using the so-called $w$-operation, we show that an integral domain $R$ is Krull if and only if for any nonzero proper $w$-ideal $I$, the Gorenstein global dimension of the $w$-factor ring $(R/I)_{w}$ is zero. Further, we obtain that an integral domain $R$ is Dedekind if and only if for any nonzero proper ideal $I$, the Gorenstein global dimension of the factor ring $R/I$ is zero.

Keywords: Krull domain, $w$-operation, $w$-factor ring, QF-ring, Gorenstein global dimension

MSC numbers: 13D05, 13F05, 13E10

Supported by: The first named author is supported by the Scientific Research Foundation of Chengdu University of Information Technology (KYTZ202015, 2022ZX001).

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