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. (2015). On characterizations of w-coherent rings. Commun. Algebra. 48(11):4681–4697).

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