- Hacettepe Journal of Mathematics and Statistics
- Vol: 52 Issue: 2
- $S$-cotorsion modules and dimensions
$S$-cotorsion modules and dimensions
Authors : Refat Abelmawla Khaled Assaad, Xiaolei Zhang
Pages : 410-419
Doi:10.15672/hujms.1093927
View : 12 | Download : 4
Publication Date : 2023-03-31
Article Type : Research
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