On characterizations of w-coherent rings II

Abstract

In this paper, we obtain that a commutative ring R is w-coherent if and only if ${\text{Hom}}_R(M,E)$ is w-flat for any absolutely pure w-module M and any injective (w-)module E, if and only if ${\text{Hom}}_R(M,E)$ is w-flat for any injective w-module M and any injective (w-)module E. To do this, we introduce the class $w-{\mathcal{E}}_w-\text{SS}-\mathcal{E}$ of all w-strictly ${\mathcal{E}}_w$-stationary modules over all injective modules $\mathcal{E}$ and show that R is w-coherent if and only if any (finitely generated) ideal of R is w-strictly ${\mathcal{E}}_w$-stationary over $\mathcal{E}.$ 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).

Keywords:

• absolutely pure w-module
• w-coherent ring
• w-flat module
• w-strictly stationary module

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 13C11
• 13C12
• 13E05

Additional information

Funding

The first author was supported by the Natural Science Foundation of Chengdu Aeronautic Polytechnic (No. 062026) and the National Natural Science Foundation of China (No. 12061001). The second author was supported by the National Natural Science Foundation of China (No. 11671283).