( Weak ) Gorenstein Global Dimension of Semiartinian Rings

Throughout the paper, all rings are commutative with identity. Let R be a ring, and let M be an R-module. As usual, we use pdR M , idR M , and fdR M to denote, respectively, the classical projective dimension, injective dimension, and flat dimension ofM. For a two-sided Noetherian ring R, Auslander and Bridger 1 introduced the G-dimension, GdimR M , for every finitely generated R-module M. They showed that GdimR M ≤ pdR M for all finitely generated R-modulesM, and equality holds if pdR M is finite. Several decades later, Enochs and Jenda 2, 3 introduced the notion of Gorenstein projective dimension G-projective dimension for short , as an extension of G dimension to modules that are not necessarily finitely generated, and the Gorenstein injective dimension G-injective dimension for short as a dual notion of Gorenstein projective dimension. Then, to complete the analogy with the classical homological dimension, Enochs et al. 4 introduced the Gorenstein flat dimension. Some references are 2–8 . Recall that an R-module M is called Gorenstein projective, if there exists an exact sequence of projective R-modules


Introduction
Throughout the paper, all rings are commutative with identity.Let R be a ring, and let M be an R-module.As usual, we use pd R M , id R M , and fd R M to denote, respectively, the classical projective dimension, injective dimension, and flat dimension of M.
For a two-sided Noetherian ring R, Auslander and Bridger 1 introduced the G-dimension, Gdim R M , for every finitely generated R-module M. They showed that Gdim R M ≤ pd R M for all finitely generated R-modules M, and equality holds if pd R M is finite.
Several decades later, Enochs and Jenda 2, 3 introduced the notion of Gorenstein projective dimension G-projective dimension for short , as an extension of G dimension to modules that are not necessarily finitely generated, and the Gorenstein injective dimension G-injective dimension for short as a dual notion of Gorenstein projective dimension.Then, to complete the analogy with the classical homological dimension, Enochs et al. 4 introduced the Gorenstein flat dimension.Some references are 2-8 .
Recall that an R-module M is called Gorenstein projective, if there exists an exact sequence of projective R-modules such that M ∼ Im P 0 −→ P 0 and such that the functor Hom R −, Q leaves P exact whenever Q is a projective R-module.The complex P is called a complete projective resolution.The Gorenstein injective R-modules are defined dually.An R-module M is called Gorenstein flat, if there exists an exact sequence of flat Rmodules: The Gorenstein projective, injective, and flat dimensions are defined in terms of resolutions and denoted by Gpd − , Gid − , and Gfd − , respectively see 6, 8, 9 .
In 5 , for any associative ring R, the authors proved the equality They called the common value of the above quantities the left Gorenstein global dimension of R and denoted it by l.Ggldim R .Similarly, they set which they called the left Gorenstein weak dimension of R. Since all rings in this paper are commutative, we drop the letter l.
Recall that an R-module M is called semiartinian, if every nonzero quotient module of M has nonzero socle.A ring R is said to be semiartinian if it is semiartinian as an R-module; see 10 .
In 11 , the authors characterized the resp., weak Gorenstein global dimension for an arbitrary associative ring.The purpose of this paper is to apply these characterizations to a commutative semiartinian rings.Hence, we prove that if R is a semiartinian commutative ring, the Gorenstein global dimension of R equals the supremum of the Gorenstein projective and injective dimension of simple R-modules Theorem 2.1 , and the weak Gorenstein global dimension of R equals the supremum of the Gorenstein flat dimensions of simple R-modules Theorem 2.7 .

Main Results
The first main result of this paper computes the Gorenstein global dimension of semiartinian rings via the Gorenstein projective and injective dimensions of simple modules.Theorem 2.1.Let R be a semiartinian ring and n a positive integer.The following conditions are equivalent: Tor n 1 R I, C 0 for all simple R-modules C, all projective R-modules P , and all injective R-modules I.
where C ranges ranges over all simple R-modules.
To prove this theorem, we need the following lemma.

2.2
Moreover, we have just proved that c ≤ n, and so a ≤ n.Accordingly, since fd R I ≤ n, we have pd R I < ∞.Hence, since b ≤ n, we get pd R I ≤ n.Consequently, the condition C2 of Lemma 2.2 is clear.As consequence, Ggldim R ≤ n, as desired.where C ranges over all simple R-modules.

Remark 2 . 3 . 3 provided R is a semiartinian ring. Corollary 2 . 4 .
From the proof of Theorem 2.1, we can easily see thatsup id R P | P is a projective R-module ≤ sup Gpd R C | C is a simple R-module , 2.Let R be a semiartinian ring with finite Gorenstein global dimension.Then, Lemma 2.2 Theorem 2.1, 11 .Let R be a ring and n a positive integer.Then, Ggldim R ≤ n if, and only if, R satisfies the following two conditions: C1 : id P ≤ n for every projective R-module P , C2 : pd I ≤ n for every injective R-module I. Proof of Theorem 2.1.1 ⇒ 2 Clear by the definition of Ggldim R . 2 ⇒ 3 By 8, Theorem 2.20 , Ext n 1 and all simple R-module C and all projective module P since Gpd R C ≤ n.Let I be an injective R-module.By 8, Theorem 2.22 , Ext n 1 Let P be a projective R-module.By 12, Lemma 4.2 2 , id R P ≤ n since Ext n 1 Hence, the condition C1 of Lemma 2.2 is clear.Let now I be an arbitrary injective R-module.By 12, Lemma 4.2 1 , fd R I ≤ n since Tor n 1 On the other hand, from 13, Theorem 7.2.5 2 and Corollary 7.2.6 1 ⇔ 2 , we have R I, C0 for all simple R-module C.