Abstract
In this paper, we obtain that a commutative ring R is w-coherent if and only if is w-flat for any absolutely pure w-module M and any injective (w-)module E, if and only if
is w-flat for any injective w-module M and any injective (w-)module E. To do this, we introduce the class
of all w-strictly
-stationary modules over all injective modules
and show that R is w-coherent if and only if any (finitely generated) ideal of R is w-strictly
-stationary over
Besides, we show that R is w-coherent if and only if any direct product of projective modules is w-flat if and only if any direct product of R is w-flat, which is a continuation of Theorem 2.14 (in Zhang, X. L., Wang, F. G., Qi, W. (Citation2015). On characterizations of w-coherent rings. Commun. Algebra. 48(11):4681–4697).