On regularly coherent modules and regularly Noetherian modules

Year 2025, Early Access, 1 - 27

Younes El Haddaoui , Hwankoo Kım , Najib Mahdou

https://doi.org/10.24330/ieja.1767099

Abstract

The concepts of regular Noetherianity and regular coherence, which extend the classical notions of Noetherian and coherent rings, have been fundamental in the study of algebraic structures. In this paper, we aim to expand these notions to the realm of module theory. Specifically, we introduce and explore weak versions of injective, flat, and projective modules, which we term as reg-injective, reg-flat, and reg-projective modules. We provide analogues of classical results and establish their properties, offering examples to illustrate modules that meet these new criteria but not their classical counterparts. Additionally, we define and study regularly Noetherian and regularly coherent modules, characterizing their properties and examining their stability under various ring constructions. Our results contribute new examples and broaden the current understanding of these algebraic concepts.

Keywords

Reg-submodule , reg-injective module , reg-flat module , reg-projective module , reg-Noetherian module , reg-coherent module

References

• D. D. Anderson, The Krull intersection theorem, Pacific J. Math., 57(1) (1975), 11-14.
• S. U. Chase, Direct products of modules, Trans. Amer. Math. Soc., 97 (1960), 457-473.
• J. Chen and N. Ding, On $n$-coherent rings, Comm. Algebra, 24(10) (1996), 3211-3216.
• M. Chhiti and S. E. Mahdou, Rings in which every regular ideal is finitely generated, Moroc. J. Algebra Geom. Appl., 2(2) (2023), 218-225.
• M. Chhiti and S. E. Mahdou, When every finitely generated regular ideal is finitely presented, Commun. Korean Math. Soc., 39(2) (2024), 363-372.
• J. Elliott, Rings, Modules, and Closure Operations, Springer Monographs in Mathematics, Springer, Cham, 2019.
• E. E. Enochs and O. M. G. Jenda, Relative Homological Algebra, De Gruyter Expositions in Mathematics, 30, Walter de Gruyter & Co., Berlin, 2000.
• S. Glaz, Commutative Coherent Rings, Lecture Notes in Mathematics, 1371, Springer-Verlag, Berlin, 1989.
• T. Y. Lam, Lectures on Modules and Rings, Graduate Texts in Mathematics, 189, Springer-Verlag, New York, 1999.
• E. Matlis, Commutative coherent rings, Canadian J. Math., 34(6) (1982), 1240-1244.
• W. Qi and X. L. Zhang, The homological properties of regular injective modules, Commun. Korean Math. Soc., 39(1) (2024), 59-69.
• B. Stenström, Coherent rings and FP-injective modules, J. London Math. Soc., 2(2) (1970), 323-329.
• B. Stenstrom, Rings of Quotients, An introduction to methods of ring theory, Die Grundlehren der Mathematischen Wissenschaften, 217, Springer-Verlag, New York-Heidelberg, 1975.
• F. G. Wang and H. Kim, Foundations of Commutative Rings and Their Modules, Algebra and Applications, 22, Springer, Singapore, 2016.
• F. G. Wang and J. L. Liao, $S$-injective modules and $S$-injective envelopes, Acta Math. Sinica (Chinese Ser.), 54(2) (2011), 271-284.
• X. L. Xiao, F. G. Wang and S. Y. Lin, The coherence study determined by regular ideals, J. Sichuan Normal Univ., 45(1) (2022), 33-40.
• X. L. Zhang, Characterizing $S$-flat modules and $S$-von Neumann regular rings by uniformity, Bull. Korean Math. Soc., 59(3) (2022), 643-657.
• X. L. Zhang, On $\tau_q$-projectivity and $\tau_q$-simplicity, (2023), arXiv:2302.04560 [math.AC].