Rings Over which All Modules are Strongly Gorenstein Projective

Abstract

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:

On (Strongly) Gorenstein (Semi)Hereditary Rings

“…On the other hand, the family {R, S } of rings satisfies the conditions of [4,Lemma 3 (2) We have to prove that R ′ × S is not strongly Gorenstein module. By [5,Corollary 3.10], there exists a Gorenstein projective R ′ -module M which is not strongly Gorenstein projective. And by, [5,Lemma 3.2], M × S is Gorenstein projective R ′ × S -module which is not strongly Gorenstein projective.…”

“…By [5,Corollary 3.10], there exists a Gorenstein projective R ′ -module M which is not strongly Gorenstein projective. And by, [5,Lemma 3.2], M × S is Gorenstein projective R ′ × S -module which is not strongly Gorenstein projective. Thus, from Proposition 2.7, R ′ × S is not strongly Gorenstein hereditary.…”

On (Strongly) Gorenstein Von Neumann Regular Rings

“…The rings 0-SG and 1-SG are already studied in [5,18] over which they are called strongly Gorenstein semi-simple and hereditary rings respectively. Clearly, by definition, every n-SG (resp., n-wSG) ring is m-SG (resp., m-wSG) whenever n ≤ m. After given some characterizations of the n-SG and n-wSG rings (see Propositions 2.1 and 2.2), we will see that for any ring R we have:…”

“…Proof. From[5, Corollary 3.9] and [3, Proposition 2.6], Ggldim(R 1 ) = Ggldim(R 2 ) = 0 and R 1 is 0-SG ring but R 2 is not. So, (1) is clear.…”

On 2-SG-semisimple rings

“…From [4, Corollary 2.8], n-SG-semisimple rings are a particular kind of quasi-Frobenius rings. In [8], it was proved that a local ring is 1-SG-semisimple if and only if it contains a unique non-trivial ideal.…”