Global Gorenstein dimensions

Abstract

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. D. BENNIS AND N. MAHDOU 2. Proofs of the main results

Strongly Gorenstein flat modules and dimensions

“…Our first result in this section shows that it coincides with the Gorenstein global dimension defined in [3]. (1) r.cot.D (R) ≤ n;…”

“…Note that Theorem 3.1 generalizes[9, Proposition 2.16] in the sense that r.SGFD (R) = 0 (i.e., every R-module is strongly Gorenstein flat) if and only if r.Ggldim (R) = 0, which means by[3, Proposition 2.6], that R is quasi-Frobenius.…”

Cartan-Eilenberg projective, injective and flat complexes

“…Then R is a QF ring since every CE injective R-complex is CE projective by (4) and Lemma 3.6. Then M is CE Gorenstein injective by [3], Proposition 2.6, and the dual version of [20], Proposition 2.15. By (4) again, M is CE projective, which gives that M is projective.…”

Gorenstein theory for n-th differential modules

“…The Gorenstein injective dimension of M, denoted by GidM, is defined dually. Moreover, the supresum of Gorenstein projective dimensions of all R-modules coincides with the supresum of Gorenstein injective dimensions of all R-modules, which is called the Gorenstein global dimension of R and is denoted by GgdR [2].…”