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 $\Tor^R_1 (M, N)$ 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 $\Ext^1_R(F,M)=0$ 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 (calssical) homological dimensions are discussed. As applications, we give a new upper bound on the global dimension of rings.
Let R be a ring and let S be a multiplicative subset of R . An R -module M is said to be a u - S -absolutely pure module if Ext R 1 N , M is u - S -torsion for any finitely presented R -module N . This paper introduces and studies the notion of S -FP-projective modules, which extends the classical notion of FP-projective modules. An R -module M is called an S -FP-projective module if Ext R 1 M , N = 0 for any u - S -absolutely pure R -module N . We also introduce the S -FP-projective dimension of a module and the global S -FP-projective dimension of a ring. Then, the relationship between the S -FP-projective dimension and other homological dimensions is discussed.
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