Descent of projectivity for locally free modules

Let R--R be the natural homomorphism from the commutative ring R into its associated von Neumann regular ring, and let M be a locally free R-module such that R?M is a projective R-module. We show that if M is either countably generated or locally finitely generated, then M is projective, and we deduce that the trace of any projective ideal is projective. These results are a consequence of a more general theorem on the descent of the Mittag-Leffler condition. The "locally free" hypothesis may be weakened to "flat" if and only if R is locally perfect.