Global Gorenstein dimensions

In this paper, we prove that the global Gorenstein projective dimension of a ring $R$ is equal to the global Gorenstein injective dimension of $R$, and that the global Gorenstein flat dimension of $R$ is smaller than the common value of the terms of this equality.


Introduction
Throughout this paper, R denotes a non-trivial associative ring with identity, and all modules are, if not specified otherwise, left R-modules. All the results, except Proposition 2.6, are formulated for left modules and the corresponding results for right modules hold as well. For an R-module M, we use pd R (M), id R (M), and fd R (M) to denote, respectively, the classical projective, injective, and flat dimension of M. We use l.gldim(R) and r.gldim (R) to denote, respectively, the classical left and right global dimension of R, and wgldim(R) to denote the weak global dimension of R. Recall that the left finitistic projective dimension of R is the quantity l.
The main result of this paper is an analog of a classical equality that is used to define the global dimension of R, see [12,Theorems 9.10]. For Noetherian rings the following theorem is proved in [4,Theorem 12 (3) l.wGgldim(R) ≤ wgldim(R). Equalities hold in (2) and (3) if wgldim(R) < ∞.
The theorem and its corollary are proved in Section 2.

Proofs of the main results
The proof use the following results: then, for a positive integer n, the following are equivalent: (2) id R (P) ≤ n for every R-module P with finite projective dimension.
The proof of the main theorem depends on the notions of strong Gorenstein projectivity and injectivity, which were introduced in [1] as follows: ). An R-module M is called strongly Gorenstein projective, if there exists an exact sequence of projective R-modules such that M Ker f and such that Hom R (−, Q) leaves the sequence P exact whenever Q is a projective R-module.
Strongly Gorenstein injective modules are defined dually. Strongly Gorenstein injective modules are characterized in similar terms.
The principal role of these modules is to characterize the Gorenstein projective and injective modules, as follows * : Theorems 2.7]). An R-module is Gorenstein projective (resp., injective) if and only if it is a direct summand of a strongly Gorenstein projective (resp., injective) R-module.
Proof of Theorem 1.1. For every integer n we need to show: We only prove the direct implication; the converse one has a dual proof. Assume first that M is strongly Gorenstein projective. By Remark 2.3 there is a short exact sequence 0 → M → P → M → 0 with P is projective. The Horseshoe Lemma, see [10, Remark page 187], gives a commutative diagram * In [1] the base ring is assumed to be commutative. However, for the result needed here, one can show easily that this assumption is not necessary.   For the case where l.Ggldim(R) = 0 or r.Ggldim(R) = 0, we have the following result which is [2, Theorem 2.2] in non-commutative setting. Recall that a ring is called quasi-Frobenius, if it is Noetherian and both left and right self-injective (see [11]). Proposition 2.6. The following are equivalent: (1) R is quasi-Frobenius; (2) l.Ggldim(R) = 0;