On characterizations of w-coherent rings

Abstract

In this article, we provide several descriptions of w-coherent rings in terms of modules. We show that a ring R is w-coherent if and only if every direct product of flat modules is w-flat. To do this, we introduce the class $w‐\mathcal{F}‐\text{ML}$ of all w-Mittag-Leffler modules with respect to all flat modules and obtain that R is w-coherent if and only if every (finitely generated) ideal is in $w‐\mathcal{F}‐\text{ML}.$ We also obtain that R is w-coherent if and only if the class of absolutely pure w-modules is closed under direct limits if and only if the class of absolutely pure w-modules is (pre)covering.

Keywords:

• Absolutely pure w-module: w-coherent ring
• w-flat module
• w-Mittag-Leffler module

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 13C11
• 13C12
• 13E05

Acknowledgement

The authors would like to thank the reviewer for many valuable suggestions on revision of this article.

Additional information

Funding

The first author was partially supported by the Natural Science Research Foundation of Chengdu Aeronautic Polytechnic (No. 062026) and the Postgraduate Fund of Sichuan Normal University (No. 201903-9). The second author was supported by NSFC (No. 11671283).