One of the main results of this paper is the characterization of the rings over which all modules are strongly Gorenstein projective. We show that these kinds of rings are very particular cases of the well-known quasi-Frobenius rings. We give examples of rings over which all modules are Gorenstein projective but not necessarily strongly Gorenstein projective.Definitions 1.1 A module M is said to be Gorenstein projective, if there exists an exact sequence of projective modules P = · · · → P 1 → P 0 → P 0 → P 1 → · · · such that M ∼ = Im(P 0 → P 0 ) and such that Hom(−, Q) leaves the sequence P exact whenever Q is a projective module.The exact sequence P is called a complete projective resolution.The Gorenstein injective modules are defined dually.Recently in [3], the authors studied a simple particular case of Gorenstein projective and injective modules, which are defined, respectively, as follows:A module M is said to be strongly Gorenstein projective, if there exists a complete projective resolution of the formThe exact sequence P is called a strongly complete projective resolution.The strongly Gorenstein injective modules are defined dually.The principal role of the strongly Gorenstein projective and injective modules is to give a simple characterization of Gorenstein projective and injective modules, respectively, as follows:Theorem 1.3 ([3], Theorem 2.7) A module is Gorenstein projective (resp., injective) if and only if it is a direct summand of a strongly Gorenstein projective (resp., injective) module.The important of this last result manifests in showing that the strongly Gorenstein projective and injective modules have simpler characterizations than their Gorenstein correspondent modules. For instance:Proposition 1.4 ([3], Proposition 2.9) A module M is strongly Gorenstein projective if and only if there exists a short exact sequence of moduleswhere P is projective, and Ext(M, Q) = 0 for any projective module Q.The aim of this paper is to investigate the two following classes of rings:
The main aim of this paper is to investigate rings over which all (finitely generated strongly) Gorenstein projective modules are projective. We consider this propriety under change of rings, and give various examples of rings with and without this propriety.
In this paper, we introduce and define the notion of almost n-perfect ring which is a generalization of almost perfect ring, and we investigate the transfer of this property to some ring extensions. Also, We give a characterization of global weak injective dimension property.
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