Let Rbe a ring with identity. Chase [5] has characterized the right coherent rings; he has shown that arbitrary products of RR are flat if and only if finitely generated submodules of free right R-modules are finitely presented. As compared to considerinq flatness of products of a projective generator, this paper investigates flatness (and related properties) of products of an injective cogenerator. This condition turns out to be intimately related to several other properties of independent interest. One is the condition that torsion-free or injective modules are flat or f-projective (the 1atter is a property stronger than fl atness); see, eg,[6],[8] and [26]. Another is the condition that finitely presented (FP) or finitely generated (FG) modules embed in free modules; see, eg,[12].

We approach these topics via torsion theory, but this paper requires a minimum of torsion theory background.(See Section 1.) Let (T, F) be an hereditary torsion theory for right R-modules, and let ERbe an injective that cogenerates (T, F). We first show (Proposition 2.1) that if Ris torsion free then arbitrary products of ERare flat if and only if every FP (relative to (T, F)), torsion-free MR