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arXiv · 2403.04676

When every S-flat module is (flat) projective

Driss Bennis·Ayoub Bouziri
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Abstract

Let R be a commutative ring with identity and S a multiplicative subset of R. The aim of this paper is to study the class of commutative rings in which every S-flat module is flat (resp., projective). An R-module M is said to be S-flat if the localization of M at S, M_S, is a flat R_S-module. Commutative rings R for which all S-flat R-modules are flat are characterized by the fact that R/Rs is a von Neumann regular ring for every s in S. While, commutative rings R for which all S-flat R-modules are projective are characterized by the following two conditions: R is perfect and the Jacobson radical J(R) of R is S-divisible. Rings satisfying these conditions are called S-perfect. Moreover, we give some examples to distinguish perfect rings, S-perfect rings, and semisimple rings. We also investigate the transfer results of the "S-perfectness" for various ring constructions, which allows the construction of more interesting examples.

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Driss Bennis, Ayoub Bouziri. 2024-03-07. When every S-flat module is (flat) projective. https://arxiv.org/abs/2403.04676

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