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Year 2023, Volume: 52 Issue: 2, 410 - 419, 31.03.2023
https://doi.org/10.15672/hujms.1093927

Abstract

References

  • [1] D.D. Anderson and T. Dumitrescu, $S$-Noetherian rings, Comm. Algebra 30, 4407- 4416, 2002.
  • [2] L. Bican, E. Bashir and E.E. Enochs, All modules have flat covers, Bull. London Math. Soc. 33, 385-390, 2001.
  • [3] N. Ding and L. Mao, The cotorsion dimension of modules and rings, in: Abelian Groups, Modules and Homological Algebra, in: Lect. Notes Pure Appl. Math. Vol.249, 217-243, Chapman and Hall, 2006.
  • [4] E.E. Enochs, Injective and flat covers, envelopes and resolvents, Israel J. Math. 39, 189-209, 1981.
  • [5] E.E. Enochs, Flat covers and flat cotorsion modules, Proc. Amer. Math. Soc. 92 (2), 179-184, 1984.
  • [6] L. Mao and N. Ding, Notes on cotorsion modules, Comm. Algebra 33, 349-360, 2005.
  • [7] J.J. Rotman, An Introduction to Homological Algebra, 2nd ed., Springer, New York, 2009.
  • [8] F. Wang and H. Kim, Foundations of Commutative Rings and Their Modules, Springer Nature Singapore Pte Ltd., Singapore, 2016.
  • [9] J. Xu, Flat covers of modules, 1st ed., Springer, Berlin, 1996.
  • [10] X.L. Zhang, Characterizing $S$-flat modules and $S$-von Neumann regular rings by uniformity, Bull. Korean Math. Soc. 59, (3), 643-657, 2022.
  • [11] X.L. Zhang, The $u$-$S$-weak global dimension of commutative rings, arXiv: 2106.00535 [math.CT].
  • [12] X.L. Zhang, $S$-absolutely pure modules, arXiv: 2108.06851 [math.CT].

$S$-cotorsion modules and dimensions

Year 2023, Volume: 52 Issue: 2, 410 - 419, 31.03.2023
https://doi.org/10.15672/hujms.1093927

Abstract

Let $R$ be a ring, $S$ a multiplicative subset of $R$. An $R$-module $M$ is said to be $u$-$S$-flat ($u$- always abbreviates uniformly) if ${\rm Tor}^R_1 (M, N)$ is $u$-$S$-torsion $R$-module for all $R$-modules $N$. In this paper, we introduce and study the concept of $S$-cotorsion module which is in some way a generalization of the notion of cotorsion module. An $R$-module $M$ is said to be $S$-cotorsion if ${\rm Ext}^1_R(F,M)=0$ for any $u$-$S$-flat module $F$. This new class of modules will be used to characterize $u$-$S$-von Neumann regular rings. Hence, we introduce the $S$-cotorsion dimensions of modules and rings. The relations between the introduced dimensions and other (classical) homological dimensions are discussed. As applications, we give a new upper bound on the global dimension of rings.

References

  • [1] D.D. Anderson and T. Dumitrescu, $S$-Noetherian rings, Comm. Algebra 30, 4407- 4416, 2002.
  • [2] L. Bican, E. Bashir and E.E. Enochs, All modules have flat covers, Bull. London Math. Soc. 33, 385-390, 2001.
  • [3] N. Ding and L. Mao, The cotorsion dimension of modules and rings, in: Abelian Groups, Modules and Homological Algebra, in: Lect. Notes Pure Appl. Math. Vol.249, 217-243, Chapman and Hall, 2006.
  • [4] E.E. Enochs, Injective and flat covers, envelopes and resolvents, Israel J. Math. 39, 189-209, 1981.
  • [5] E.E. Enochs, Flat covers and flat cotorsion modules, Proc. Amer. Math. Soc. 92 (2), 179-184, 1984.
  • [6] L. Mao and N. Ding, Notes on cotorsion modules, Comm. Algebra 33, 349-360, 2005.
  • [7] J.J. Rotman, An Introduction to Homological Algebra, 2nd ed., Springer, New York, 2009.
  • [8] F. Wang and H. Kim, Foundations of Commutative Rings and Their Modules, Springer Nature Singapore Pte Ltd., Singapore, 2016.
  • [9] J. Xu, Flat covers of modules, 1st ed., Springer, Berlin, 1996.
  • [10] X.L. Zhang, Characterizing $S$-flat modules and $S$-von Neumann regular rings by uniformity, Bull. Korean Math. Soc. 59, (3), 643-657, 2022.
  • [11] X.L. Zhang, The $u$-$S$-weak global dimension of commutative rings, arXiv: 2106.00535 [math.CT].
  • [12] X.L. Zhang, $S$-absolutely pure modules, arXiv: 2108.06851 [math.CT].
There are 12 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Refat Abelmawla Khaled Assaad 0000-0001-8616-0176

Xiaolei Zhang 0000-0003-0018-4180

Publication Date March 31, 2023
Published in Issue Year 2023 Volume: 52 Issue: 2

Cite

APA Assaad, R. A. K., & Zhang, X. (2023). $S$-cotorsion modules and dimensions. Hacettepe Journal of Mathematics and Statistics, 52(2), 410-419. https://doi.org/10.15672/hujms.1093927
AMA Assaad RAK, Zhang X. $S$-cotorsion modules and dimensions. Hacettepe Journal of Mathematics and Statistics. March 2023;52(2):410-419. doi:10.15672/hujms.1093927
Chicago Assaad, Refat Abelmawla Khaled, and Xiaolei Zhang. “$S$-Cotorsion Modules and Dimensions”. Hacettepe Journal of Mathematics and Statistics 52, no. 2 (March 2023): 410-19. https://doi.org/10.15672/hujms.1093927.
EndNote Assaad RAK, Zhang X (March 1, 2023) $S$-cotorsion modules and dimensions. Hacettepe Journal of Mathematics and Statistics 52 2 410–419.
IEEE R. A. K. Assaad and X. Zhang, “$S$-cotorsion modules and dimensions”, Hacettepe Journal of Mathematics and Statistics, vol. 52, no. 2, pp. 410–419, 2023, doi: 10.15672/hujms.1093927.
ISNAD Assaad, Refat Abelmawla Khaled - Zhang, Xiaolei. “$S$-Cotorsion Modules and Dimensions”. Hacettepe Journal of Mathematics and Statistics 52/2 (March 2023), 410-419. https://doi.org/10.15672/hujms.1093927.
JAMA Assaad RAK, Zhang X. $S$-cotorsion modules and dimensions. Hacettepe Journal of Mathematics and Statistics. 2023;52:410–419.
MLA Assaad, Refat Abelmawla Khaled and Xiaolei Zhang. “$S$-Cotorsion Modules and Dimensions”. Hacettepe Journal of Mathematics and Statistics, vol. 52, no. 2, 2023, pp. 410-9, doi:10.15672/hujms.1093927.
Vancouver Assaad RAK, Zhang X. $S$-cotorsion modules and dimensions. Hacettepe Journal of Mathematics and Statistics. 2023;52(2):410-9.