Article
| Title: | A note on $S$-flat preenvelopes (English) |
| Author: | Zhang, Xiaolei |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 75 |
| Issue: | 3 |
| Year: | 2025 |
| Pages: | 785-792 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | 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. (English) |
| Keyword: | $S$-coherent ring |
| Keyword: | $S$-flat module |
| Keyword: | $S$-flat preenvelope |
| MSC: | 13C11 |
| DOI: | 10.21136/CMJ.2025.0352-24 |
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| Date available: | 2025-09-19T11:47:05Z |
| Last updated: | 2025-09-22 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/153051 |
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| Reference: | [1] Anderson, D. D., Dumitrescu, T.: $S$-Noetherian rings.Commun. Algebra 30 (2002), 4407-4416. Zbl 1060.13007, MR 1936480, 10.1081/AGB-120013328 |
| Reference: | [2] Anderson, D. D., Winders, M.: Idealization of a module.J. Commut. Algebra 1 (2009), 3-56. Zbl 1194.13002, MR 2462381, 10.1216/JCA-2009-1-1-3 |
| Reference: | [3] Baeck, J.: $S$-injective modules.Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM 118 (2024), Article ID 20, 20 pages. Zbl 1540.16005, MR 4662890, 10.1007/s13398-023-01514-7 |
| Reference: | [4] Bennis, D., Bouziri, A.: $S$-flat cotorsion pair.Bull. Korean Math. Soc. 62 (2025), 237-250. Zbl 08014187, MR 4857929, 10.4134/BKMS.b240152 |
| Reference: | [5] Bennis, D., Hajoui, M. El: On $S$-coherence.J. Korean Math. Soc. 55 (2018), 1499-1512. Zbl 1405.13035, MR 3883493, 10.4134/JKMS.j170797 |
| Reference: | [6] Chase, S. U.: Direct products of modules.Trans. Am. Math. Soc. 97 (1960), 457-473. Zbl 0100.26602, MR 0120260, 10.1090/S0002-9947-1960-0120260-3 |
| Reference: | [7] Dai, G., Ding, N.: Coherent rings and absolutely pure covers.Commun. Algebra 46 (2018), 1267-1271. Zbl 1442.16003, MR 3780239, 10.1080/00927872.2017.1344693 |
| Reference: | [8] Dai, G., Ding, N.: Coherent rings and absolutely pure precovers.Commun. Algebra 47 (2019), 4743-4748. Zbl 1470.16008, MR 3991048, 10.1080/00927872.2019.1595637 |
| Reference: | [9] Enochs, E. E.: Injective and flat covers, envelopes and resolvents.Isr. J. Math. 39 (1981), 189-209. Zbl 0464.16019, MR 0636889, 10.1007/BF02760849 |
| Reference: | [10] Enochs, E. E., Jenda, O. M. G.: Relative Homological Algebra. I.de Gruyter Expositions in Mathematics 30. Walter de Gruyter, Berlin (2011). Zbl 1238.13001, MR 2857612, 10.1515/9783110215212 |
| Reference: | [11] Fuchs, L., Salce, L.: Modules over non-Noetherian Domains.Mathematical Surveys and Monographs 84. AMS, Providence (2001). Zbl 0973.13001, MR 1794715, 10.1090/surv/084 |
| Reference: | [12] Qi, W., Zhang, X., Zhao, W.: New characterizations of $S$-coherent rings.J. Algebra Appl. 22 (2023), Article ID 2350078, 14 pages. Zbl 1516.13008, MR 4553213, 10.1142/S0219498823500780 |
| Reference: | [13] Stenström, B.: Coherent rings and FP-injective modules.J. Lond. Math. Soc., II. Ser. 2 (1970), 323-329. Zbl 0194.06602, MR 0258888, 10.1112/jlms/s2-2.2.323 |
| Reference: | [14] Wang, F., Kim, H.: Foundations of Commutative Rings and Their Modules.Algebra and Applications 22. Springer, Singapore (2016). Zbl 1367.13001, MR 3587977, 10.1007/978-981-10-3337-7 |
| Reference: | [15] Zhang, X.: The $u$-$S$-weak global dimensions of commutative rings.Commun. Korean Math. Soc. 38 (2023), 97-112. Zbl 1525.13019, MR 4542624, 10.4134/CKMS.c220016 |
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