The cotorsion pair generated by the class of flat Mittag-Leffler modules

Abstract

Let R be an associative ring with unit and denote by FM the class of all flat and Mittag-Leffler left R-modules. In [4] it is proved that, if R is countable, the orthogonal class of FM consists of all cotorsion modules. In this note we extend this result to the class of all rings R satisfying that each flat left R-module is filtered by totally ordered direct limits of projective modules. This class of rings contains all countable, left perfect and discrete valuation domains. Moreover, assuming that there do not exist inaccessible cardinals, we obtain that, over these rings, all flat left R-modules have finite projective dimension.