Czechoslovak Mathematical Journal

Czechoslovak Mathematical Journal, first online, pp. 1-8

A note on $S$-flat preenvelopes

Xiaolei Zhang

Received August 20, 2024. Published online May 19, 2025.

Abstract: We investigate the notion of $S$-flat preenvelopes of modules. In particular, we give an example that a ring $R$ being coherent does not imply that every $R$-module has an $S$-flat preenvelope, giving a negative answer to the question proposed by D. Bennis and A. Bouziri (2025). Besides, we also show that a ring $R_S$ being coherent also does not imply that $R$ is an $S$-coherent ring in general.

Keywords: $S$-coherent ring; $S$-flat module; $S$-flat preenvelope

Classification MSC: 13C11

DOI: 10.21136/CMJ.2025.0352-24

PDF available at: Institute of Mathematics CAS

References:
[1] D. D. Anderson, T. Dumitrescu: $S$-Noetherian rings. Commun. Algebra 30 (2002), 4407-4416. DOI 10.1081/AGB-120013328 | MR 1936480 | Zbl 1060.13007
[2] D. D. Anderson, M. Winders: Idealization of a module. J. Commut. Algebra 1 (2009), 3-56. DOI 10.1216/JCA-2009-1-1-3 | MR 2462381 | Zbl 1194.13002
[3] J. Baeck: $S$-injective modules. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM 118 (2024), Article ID 20, 20 pages. DOI 10.1007/s13398-023-01514-7 | MR 4662890 | Zbl 1540.16005
[4] D. Bennis, A. Bouziri: $S$-flat cotorsion pair. Bull. Korean Math. Soc. 62 (2025), 237-250. DOI 10.4134/BKMS.b240152 | MR 4857929 | Zbl 08014187
[5] D. Bennis, M. El Hajoui: On $S$-coherence. J. Korean Math. Soc. 55 (2018), 1499-1512. DOI 10.4134/JKMS.j170797 | MR 3883493 | Zbl 1405.13035
[6] S. U. Chase: Direct products of modules. Trans. Am. Math. Soc. 97 (1960), 457-473. DOI 10.1090/S0002-9947-1960-0120260-3 | MR 0120260 | Zbl 0100.26602
[7] G. Dai, N. Ding: Coherent rings and absolutely pure covers. Commun. Algebra 46 (2018), 1267-1271. DOI 10.1080/00927872.2017.1344693 | MR 3780239 | Zbl 1442.16003
[8] G. Dai, N. Ding: Coherent rings and absolutely pure precovers. Commun. Algebra 47 (2019), 4743-4748. DOI 10.1080/00927872.2019.1595637 | MR 3991048 | Zbl 1470.16008
[9] E. E. Enochs: Injective and flat covers, envelopes and resolvents. Isr. J. Math. 39 (1981), 189-209. DOI 10.1007/BF02760849 | MR 0636889 | Zbl 0464.16019
[10] E. E. Enochs, O. M. G. Jenda: Relative Homological Algebra. I. de Gruyter Expositions in Mathematics 30. Walter de Gruyter, Berlin (2011). DOI 10.1515/9783110215212 | MR 2857612 | Zbl 1238.13001
[11] L. Fuchs, L. Salce: Modules over non-Noetherian Domains. Mathematical Surveys and Monographs 84. AMS, Providence (2001). DOI 10.1090/surv/084 | MR 1794715 | Zbl 0973.13001
[12] W. Qi, X. Zhang, W. Zhao: New characterizations of $S$-coherent rings. J. Algebra Appl. 22 (2023), Article ID 2350078, 14 pages. DOI 10.1142/S0219498823500780 | MR 4553213 | Zbl 1516.13008
[13] B. Stenström: Coherent rings and FP-injective modules. J. Lond. Math. Soc., II. Ser. 2 (1970), 323-329. DOI 10.1112/jlms/s2-2.2.323 | MR 0258888 | Zbl 0194.06602
[14] F. Wang, H. Kim: Foundations of Commutative Rings and Their Modules. Algebra and Applications 22. Springer, Singapore (2016). DOI 10.1007/978-981-10-3337-7 | MR 3587977 | Zbl 1367.13001
[15] X. Zhang: The $u$-$S$-weak global dimensions of commutative rings. Commun. Korean Math. Soc. 38 (2023), 97-112. DOI 10.4134/CKMS.c220016 | MR 4542624 | Zbl 1525.13019

Affiliations: Xiaolei Zhang, School of Mathematics and Statistics, Tianshui Normal University, South Xihe Road, Qinzhou District, Tianshui 741001, Gansu Province, P. R. China, School of Mathematics and Statistics, Shandong University of Technology, 266 Xincun W Rd, Zhangdian District, Zibo 255049, Shandong, P. R. China, e-mail: zxlrghj@163.com

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