Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

THE HOMOLOGICAL PROPERTIES OF REGULAR INJECTIVE MODULES

  • Wei Qi (School of Mathematics and Statistics Shandong University of Technology) ;
  • Xiaolei Zhang (School of Mathematics and Statistics Shandong University of Technology)
  • Received : 2023.04.09
  • Accepted : 2023.07.20
  • Published : 2024.01.31
Download PDF
⟨ Previous Next ⟩

Abstract

Let R be a commutative ring. An R-module E is said to be regular injective provided that Ext1R(R/I, E) = 0 for any regular ideal I of R. We first show that the class of regular injective modules have the hereditary property, and then introduce and study the regular injective dimension of modules and regular global dimension of rings. Finally, we give some homological characterizations of total rings of quotients and Dedekind rings.

Keywords

Acknowledgement

The authors are very grateful to the reviewers for their suggestions on the article. The first author was supported by National Natural Science Foundation of China (No. 12201361).

References

  1. D. D. Anderson and M. Winders, Idealization of a module, J. Commut. Algebra 1 (2009), no. 1, 3-56. https://doi.org/10.1216/JCA-2009-1-1-3
  2. H. Cartan and S. Eilenberg, Homological Algebra, Princeton Univ. Press, Princeton, NJ, 1956.
  3. R. F. Damiano, Coflat rings and modules, Pacific J. Math. 81 (1979), no. 2, 349-369. http://projecteuclid.org/euclid.pjm/1102785279 102785279
  4. A. J. De Jong et al., The Stacks project, https://stacks.math.columbia.edu/
  5. L. Fuchs and L. Salce, Modules over non-Noetherian domains, Mathematical Surveys and Monographs, 84, Amer. Math. Soc., Providence, RI, 2001. https://doi.org/10.1090/surv/084
  6. J. Huckaba, Commutative rings with zero divisors, Monographs and Textbooks in Pure and Applied Mathematics, 117, Marcel Dekker, Inc., New York, 1988.
  7. B. H. Maddox, Absolutely pure modules, Proc. Amer. Math. Soc. 18 (1967), 155-158. https://doi.org/10.2307/2035245
  8. C. Megibben, Absolutely pure modules, Proc. Amer. Math. Soc. 26 (1970), 561-566. https://doi.org/10.2307/2037108
  9. B. L. Osofsky, Homological dimensions of modules, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 12, Amer. Math. Soc., Providence, RI, 1973.
  10. F. G. Wang and H. Kim, Foundations of commutative rings and their modules, Algebra and Applications, 22, Springer, Singapore, 2016. https://doi.org/10.1007/978-981-10-3337-7
  11. F. G. Wang and J. L. Liao, S-injective modules and S-injective envelopes, Acta Math. Sinica (Chinese Ser.) 54 (2011), no. 2, 271-284.
  12. M. Y. Wang and G. Zhao, On maximal injectivity, Acta Math. Sin. (Engl. Ser.) 21 (2005), no. 6, 1451-1458. https://doi.org/10.1007/s10114-005-0599-0
  13. X. Y. Yang, Generalized Noetherian Property of Rings and Modules, Lanzhou: North-west Normal University, 2006.