$S$-cotorsion modules and dimensions

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.

Keywords

• S-cotorsion module
• u-S-flat module
• S-cotorsion dimension
• global S-cotorsion dimension

References

1. [1] D.D. Anderson and T. Dumitrescu, $S$-Noetherian rings, Comm. Algebra 30, 4407- 4416, 2002.
2. [2] L. Bican, E. Bashir and E.E. Enochs, All modules have flat covers, Bull. London Math. Soc. 33, 385-390, 2001.
3. [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. [4] E.E. Enochs, Injective and flat covers, envelopes and resolvents, Israel J. Math. 39, 189-209, 1981.
5. [5] E.E. Enochs, Flat covers and flat cotorsion modules, Proc. Amer. Math. Soc. 92 (2), 179-184, 1984.
6. [6] L. Mao and N. Ding, Notes on cotorsion modules, Comm. Algebra 33, 349-360, 2005.
7. [7] J.J. Rotman, An Introduction to Homological Algebra, 2nd ed., Springer, New York, 2009.
8. [8] F. Wang and H. Kim, Foundations of Commutative Rings and Their Modules, Springer Nature Singapore Pte Ltd., Singapore, 2016.

9. [9] J. Xu, Flat covers of modules, 1st ed., Springer, Berlin, 1996.
10. [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. [11] X.L. Zhang, The $u$-$S$-weak global dimension of commutative rings, arXiv: 2106.00535 [math.CT].
12. [12] X.L. Zhang, $S$-absolutely pure modules, arXiv: 2108.06851 [math.CT].