$S$-cotorsion modules and dimensions

Authors : Refat Abelmawla Khaled ASSAAD, Xiaolei ZHANG

Pages : 410-419

Doi:10.15672/hujms.1093927

View : 22 | Download : 11

Publication Date : 2023-03-31

Article Type : Research Paper

Abstract :Let $R$ be a ring, $S$ a multiplicative subset of $R$. An $R$-module $M$ is said to be $u$-$S$-flat insert ignore into journalissuearticles values($u$- always abbreviates uniformly); if ${\\rm Tor}^R_1 insert ignore into journalissuearticles values(M, N);$ 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 ${\\rm Ext}^1_Rinsert ignore into journalissuearticles values(F,M);=0$ 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 insert ignore into journalissuearticles values(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