On characterizations of w-coherent rings II. Issue 9 (2nd September 2021)

• Record Type: Journal Article
• Authors: Zhang, Xiaolei; Wang, Fanggui
• Is Part Of: Communications in algebra. Volume 49:Issue 9(2021)
• Page Start: 3926
• Page End: 3940
• Publication Date: 2021-09-02
• Subjects: absolutely pure w-module -- w-coherent ring -- w-flat module -- w-strictly stationary module; 13C11 -- 13C12 -- 13E05; Algebra -- Periodicals; 512.005
• DOI: 10.1080/00927872.2021.1909607
• Languages: English
• ISSNs: 0092-7872

Abstract

Abstract: In this paper, we obtain that a commutative ring R is w -coherent if and only if Hom R ( M, E ) is w -flat for any absolutely pure w -module M and any injective ( w -)module E, if and only if 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 − E w − SS − E of all w -strictly E w -stationary modules over all injective modules E and show that R is w -coherent if and only if any (finitely generated) ideal of R is w -strictly E w -stationary over 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).