Products of flat modules and global dimension relative to \(\mathcal{F}\)-Mittag-Leffler modules (Q2821715)

Let \(R\) be a ring with identity and \(\mathcal{X}\) be any class of right \(R\)-modules. A left \(R\)-module \(M\) is called \(\mathcal{X}\)-Mittag-Leffler if for each family of right modules \(\{F_i\mid i\in I\}\) belonging to \(\mathcal{X}\), the canonical map from \((\prod_{i\in I}F_i)\otimes M\) to \(\prod_{i\in I}F_i\otimes M\) is monic.NEWLINENEWLINEIt is shown in this paper 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. To prove this result, the author first shows that 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 \(\mathcal{X}\).