Narrow Results

This paper focuses on the study of $\varphi$-weak global dimensions in the context of polynomial rings and $\varphi$-Prüfer rings. We explore new properties of these dimensions and extend the Hilbert syzygy theorem to $\varphi$-weak global dimensions of rings. We also determine the $\varphi$-weak global dimension for certain types of $\varphi$-Prüfer rings. Key concepts such as $\varphi$-flat modules, $\varphi$-injective modules, and $\varphi$-torsion modules are discussed, along with their hereditary properties in PN-rings. This paper includes several theorems and lemmas that provide insights into the $\varphi$-weak global dimensions and their implications in the field of ring theory.

In this paper, we will introduce and study the homological dimensions defined in the context of commutative rings with prime nilradical. So all rings considered in this paper are commutative with identity and with prime nilradical. We will introduce a new class of modules which are called $\varphi$-u-projective which generalizes the projectivity in the classical case and which is different from those introduced by the authors of [Y. Pu, M. Wang and W. Zhao, On nonnil-commutative diagrams and nonnil-projective modules, Commun. Algebra, doi:10.1080/00927872.2021.2021223; W. Zhao, On $\varphi$-exact sequence and $\varphi$-projective module, J. Korean Math. 58(6) (2021) 1513–1528]. Using the notion of $\varphi$-flatness introduced and studied by the authors of [G. H. Tang, F. G. Wang and W. Zhao, On $\varphi$-Von Neumann regular rings, J. Korean Math. Soc. 50(1) (2013) 219–229] and the nonnil-injectivity studied by the authors of [W. Qi and X. L. Zhang, Some Remarks on $\varphi$-Dedekind rings and $\varphi$-Prüfer rings, preprint (2022), arXiv:2103.08278v2 [math.AC]; X. Y. Yang, Generalized Noetherian Property of Rings and Modules (Northwest Normal University Library, Lanzhou, 2006); X. L. Zhang, Strongly $\varphi$-flat modules, strongly nonnil-injective modules and their homological dimensions, preprint (2022), https://arxiv.org/abs/2211.14681; X. L. Zhang and W. Zhao, On Nonnil-injective modules, J. Sichuan Normal Univ. 42(6) (2009) 808–815; W. Zhao, Homological theory over NP-rings and its applications (Sichuan Normal University, Chengdu, 2013)], we will introduce the $\varphi$-injective dimension, $\varphi$-projective dimension and $\varphi$-flat dimension for modules, and also the $\varphi$-(weak) global dimension of rings. Then, using these dimensions, we characterize several $\varphi$-rings ($\varphi$-Prüfer, $\varphi$-chained, $\varphi$-von Neumann, etc). Finally, we study the $\varphi$-(weak) global dimension of the trivial ring extensions defined by some conditions.