… First of all, we will study the general theory of envelopes and … Cohen-Macaulay rings and
Gorenstein rings by the dual Bass … every finitely generated right R-module is finitely presented. …

… A finitely generated R-module ω R is a canonical module for R if there is an s ∈ S such
that … If f: ⊕ i∈I F i → ⊕ i∈I F i satisfies f ˆ φ = φ, then the diagram is commutative. So f is an …

… Throughout this paper, all rings are nonzero Gorenstein rings, ie, … have the definitions of a
C -preenvelope and a C -envelope. … sum of finitely generated Gorenstein projective modules. …

… will see that all of the rings A = K[x]/(xk) are graded Gorenstein K-… Gorenstein projective
model structure on A-Mod always lifts to a ‘… We say it has enough injectives if it satisfies the dual …

… Finally, we prove a Gorenstein analogue of Costa’s first … injective envelope by [47, Theorem
3.8], and every left R-module has … where each Fi is finitely generated and free, from which we …

… us to introduce injective modules, injective envelope, and Matlis duality. … modules, and
Gorenstein rings are characterized by the fact that every finitely generated module has finite …

… the class of Gorenstein flat modules over a right coherent rings, … The dual notion is that of
(pre-)envelopes: if E is a class, the … we assumed S was finitely generated and projective over R, …

… is an Iwanaga–Gorenstein ring with the injective envelope of \(… In this paper, we consider the
rings satisfying the Auslander … projective resolution by finitely generated projective modules. …

… that every module over any ring has an injective envelope as … 1-Gorenstein domains (and
general 1-Gorenstein rings), we … fqp-ring by subjecting all finitely generated ideals to any one …

… In Section i we consider, for a finitely generated module M … ring of a Gorenstein module G
over a local ring A necessarily … and let E be the A/xA injective envelope of k. From remarks (3.3) …