Korean J. Math. Vol. 33 No. 3 (2025) pp.261-271
DOI: https://doi.org/10.11568/kjm.2025.33.3.261

On the weak global dimension of a subclass of Pr\"ufer non-coherent rings

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Published: Sep 30, 2025
Keywords:
($\phi$-)Pr\"ufer ring, $\phi$-$D$-ring, weak global dimension
Mathematics Subject Classification:
13A15, 13C05, 13C10, 13C11, 13C12, 13D05, 13F05.

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Younes El Haddaoui
Department of Mathematics, University S.M. Ben Abdellah, Fez, Morocco
Hwankoo Kim
Division of Computer Engineering, Hoseo University, Asan, Republic of Korea
Najib Mahdou
Department of Mathematics, University S.M. Ben Abdellah, Fez, Morocco

Abstract

It is known that if $R$ is a coherent Pr\"ufer ring, which is necessarily a Gaussian ring, then its weak global dimension w.gl.dim($R$) must be $0$, $1$, or $\infty$.
In this paper, we investigate the possible values of the weak global dimension for a broader class of Pr\"ufer rings that are not necessarily coherent. Our analysis employs four conceptually distinct proofs, each relying on different homological techniques, including localization at the nilradical, finitistic projective dimension, and flatness properties. The results extend the classical framework to a non-coherent setting by incorporating the effective $\mathcal{H}^D$ framework, which serves as a surrogate for coherence in controlling homological dimensions. This work aims to deepen the understanding of the weak global dimension in the context of non-coherent Pr\"ufer rings and provide a unified perspective on its behavior.



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