TY - JOUR

T1 - Generalized Gorenstein Modules

AU - Iacob, Alina

N1 - Publisher Copyright:
© 2022 Academy of Mathematics and Systems Science, Chinese Academy of Sciences.

PY - 2022/12/1

Y1 - 2022/12/1

N2 - We introduce a generalization of the Gorenstein injective modules: the Gorenstein FPn-injective modules (denoted by GIn). They are the cycles of the exact complexes of injective modules that remain exact when we apply a functor Hom(A,-), with A any FPn-injective module. Thus, GI0 is the class of classical Gorenstein injective modules, and GI1 is the class of Ding injective modules. We prove that over any ring R, for any n≥2, the class GIn is the right half of a perfect cotorsion pair, and therefore it is an enveloping class. For n=1 we show that GI1 (i.e., the Ding injectives) forms the right half of a hereditary cotorsion pair. If moreover the ring R is coherent, then the Ding injective modules form an enveloping class. We also define the dual notion, that of Gorenstein FPn-projectives (denoted by GIn). They generalize the Ding projective modules, and so, the Gorenstein projective modules. We prove that for anyn≥2 the class GPn is the left half of a complete hereditary cotorsion pair, and therefore it is special precovering.

AB - We introduce a generalization of the Gorenstein injective modules: the Gorenstein FPn-injective modules (denoted by GIn). They are the cycles of the exact complexes of injective modules that remain exact when we apply a functor Hom(A,-), with A any FPn-injective module. Thus, GI0 is the class of classical Gorenstein injective modules, and GI1 is the class of Ding injective modules. We prove that over any ring R, for any n≥2, the class GIn is the right half of a perfect cotorsion pair, and therefore it is an enveloping class. For n=1 we show that GI1 (i.e., the Ding injectives) forms the right half of a hereditary cotorsion pair. If moreover the ring R is coherent, then the Ding injective modules form an enveloping class. We also define the dual notion, that of Gorenstein FPn-projectives (denoted by GIn). They generalize the Ding projective modules, and so, the Gorenstein projective modules. We prove that for anyn≥2 the class GPn is the left half of a complete hereditary cotorsion pair, and therefore it is special precovering.

KW - Ding injective modules

KW - Ding projective modules

KW - Gorenstein FP n -injective modules

KW - Gorenstein FP n -projective modules

UR - http://www.scopus.com/inward/record.url?scp=85143861333&partnerID=8YFLogxK

U2 - 10.1142/S1005386722000463

DO - 10.1142/S1005386722000463

M3 - Article

AN - SCOPUS:85143861333

SN - 1005-3867

VL - 29

SP - 651

EP - 662

JO - Algebra Colloquium

JF - Algebra Colloquium

IS - 4

ER -