  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.
