Product of flat modules and global dimension relative to $\mathcal F$-Mittag-Leffler modules

Abstract: Let $R$ be any ring. We prove that all direct products of flat right $R$-modules have finite flat dimension if and only if each finitely generated left ideal of $R$ has finite projective dimension relative to the class of all $\mathcal F$-Mittag-Leffler left $R$-modules, where $\mathcal F$ is the class of all flat right $R$-modules. In order to prove this theorem, we obtain a general result concerning global relative dimension. Namely, if $\mathcal X$ is any class of left $R$-modules closed under filtrations that contains all projective modules, then $R$ has finite left global projective dimension relative to $\mathcal X$ if and only if each left ideal of $R$ has finite projective dimension relative to $\mathcal X$. This result contains, as particular cases, the well known results concerning the classical left global, weak and Gorenstein global dimensions.