When every S-flat module is (flat) projective

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.

KEYWORDS:

• Perfect rings
• S-almost perfect rings
• S-perfect rings
• S-flat modules
• semisimple rings

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 13A05
• 13D02
• 13C99

Acknowledgments

The authors wish to express their gratitude to the referee for the careful critical reading of the manuscript and for his/her valuable comments.

Notes

1 One can see that the fact that every R-module is S-flat is equivalent to saying that R_S is a von Neumann regular ring. So there is no reason to call such a ring S-von Neumann regular.