TY - JOUR

T1 - Weakly Ding injective modules and complexes

AU - Iacob, Alina

N1 - Publisher Copyright:
© 2023 Taylor & Francis Group, LLC.

PY - 2023

Y1 - 2023

N2 - We consider two classes of modules over coherent rings: the Gorenstein FP-injective modules, and the weakly Ding injective modules. The Gorenstein FP-injective modules are the cycles of the exact complexes of FP-injective modules. The weakly Ding injective modules are the cycles of the exact complexes of FP-injective modules that stay exact when applying a functor (Formula presented.) for any FP-injective module A. We prove that the class of Gorenstein FP-injective modules is both covering and preenveloping over any (left) coherent ring with the property that every injective module has finite flat dimension. We also prove that, over the same type of rings, the class of weakly Ding injectives, (Formula presented.), is preenveloping in (Formula presented.). If, moreover, (Formula presented.) is closed under extensions, then (Formula presented.) is a hereditary cotorsion pair. In particular, this is the case when R is a Ding Chen ring, and (Formula presented.) is closed under extensions. We show that if R is a Ding Chen ring such that every FP-injective module has finite projective dimension, then the two classes of modules coincide. Consequently, the class of weakly Ding injective modules is also covering in this case. In the last part of the paper we prove some analogue results for Gorenstein FP-injective complexes and for weakly Ding injective complexes.

AB - We consider two classes of modules over coherent rings: the Gorenstein FP-injective modules, and the weakly Ding injective modules. The Gorenstein FP-injective modules are the cycles of the exact complexes of FP-injective modules. The weakly Ding injective modules are the cycles of the exact complexes of FP-injective modules that stay exact when applying a functor (Formula presented.) for any FP-injective module A. We prove that the class of Gorenstein FP-injective modules is both covering and preenveloping over any (left) coherent ring with the property that every injective module has finite flat dimension. We also prove that, over the same type of rings, the class of weakly Ding injectives, (Formula presented.), is preenveloping in (Formula presented.). If, moreover, (Formula presented.) is closed under extensions, then (Formula presented.) is a hereditary cotorsion pair. In particular, this is the case when R is a Ding Chen ring, and (Formula presented.) is closed under extensions. We show that if R is a Ding Chen ring such that every FP-injective module has finite projective dimension, then the two classes of modules coincide. Consequently, the class of weakly Ding injective modules is also covering in this case. In the last part of the paper we prove some analogue results for Gorenstein FP-injective complexes and for weakly Ding injective complexes.

KW - Ding injective complexes

KW - Ding injective modules

UR - http://www.scopus.com/inward/record.url?scp=85161853070&partnerID=8YFLogxK

U2 - 10.1080/00927872.2023.2222411

DO - 10.1080/00927872.2023.2222411

M3 - Article

AN - SCOPUS:85161853070

SN - 0092-7872

VL - 51

SP - 4899

EP - 4912

JO - Communications in Algebra

JF - Communications in Algebra

IS - 12

ER -