ISRN Algebra

VolumeΒ 2011, Article IDΒ 497831, 6 pages

http://dx.doi.org/10.5402/2011/497831

## (Weak) Gorenstein Global Dimension of Semiartinian Rings

Department of Mathematics, Faculty of Science and Technology of Fez, University S.M. Ben Abdellah Fez, Morocco

Received 18 April 2011; Accepted 20 May 2011

Academic Editors: A.Β Jaballah and D.Β Kressner

Copyright Β© 2011 Mohammed Tamekkante and Mohamed Chhiti. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We prove that if is a semiartinian commutative ring, the Gorenstein global dimension of equals the supremum of the Gorenstein projective and injective dimensions of simple -modules, and the weak Gorenstein global dimension of equals the supremum of the Gorenstein flat dimensions of simple -modules.

#### 1. Introduction

Throughout the paper, all rings are commutative with identity. Let be a ring, and let be an -module. As usual, we use , , and to denote, respectively, the classical projective dimension, injective dimension, and flat dimension of .

For a two-sided Noetherian ring , Auslander and Bridger [1] introduced the -dimension, , for every finitely generated -module . They showed that for all finitely generated -modules , and equality holds if is finite.

Several decades later, Enochs and Jenda [2, 3] introduced the notion of Gorenstein projective dimension (-projective dimension for short), as an extension of dimension to modules that are not necessarily finitely generated, and the Gorenstein injective dimension (-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 [2β8].

Recall that an -module is called Gorenstein projective, if there exists an exact sequence of projective -modules

such that and such that the functor leaves exact whenever is a projective -module. The complex is called a complete projective resolution.

The Gorenstein injective -modules are defined dually.

An -module is called Gorenstein flat, if there exists an exact sequence of flat -modules:

such that and such that the functor leaves exact whenever is a right injective -module. The complex is called a complete flat resolution.

The Gorenstein projective, injective, and flat dimensions are defined in terms of resolutions and denoted by Gpd(β), Gid(β), and Gfd(β), respectively (see [6, 8, 9]).

In [5], for any associative ring , the authors proved the equality They called the common value of the above quantities the left Gorenstein global dimension of and denoted it by . Similarly, they set which they called the left Gorenstein weak dimension of . Since all rings in this paper are commutative, we drop the letter .

Recall that an -module is called semiartinian, if every nonzero quotient module of has nonzero socle. A ring is said to be semiartinian if it is semiartinian as an -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 is a semiartinian commutative ring, the Gorenstein global dimension of equals the supremum of the Gorenstein projective and injective dimension of simple -modules (Theorem 2.1), and the weak Gorenstein global dimension of equals the supremum of the Gorenstein flat dimensions of simple -modules (Theorem 2.7).

#### 2. 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 be a semiartinian ring and a positive integer. The following conditions are equivalent: *(1)*,
*(2)* where ranges ranges over all simple -modules, *(3)* for all simple -modules , all projective -modules , and all injective -modules . ** Consequently,
**
where ranges ranges over all simple -modules.*

To prove this theorem, we need the following lemma.

Lemma 2.2 (Theorem 2.1, [11]). *Let be a ring and a positive integer. Then, if, and only if, satisfies the following two conditions: *(C1):* for every projective -module , *(C2):* for every injective -module .*

*Proof of Theorem 2.1. *(1)*β*(2) Clear by the definition of .

(2)*β*(3) By [8, Theoremββ2.20], for all and all simple -module and all projective module since . Let be an injective -module. By [8, Theoremββ2.22], for all and all simple -module since . Then, by [12, Lemmaββ3.1(1)], for every simple -module .

(3)*β*(1) Let be a projective -module. By [12, Lemmaββ4.2(2)], since for all simple -modules . Hence, the condition (C1) of Lemma 2.2 is clear. Let now be an arbitrary injective -module. By [12, Lemmaββ4.2(1)], since for all simple -module . On the other hand, from [13, Theoremββ7.2.5(2) and Corollaryββ7.2.6(1)(2)], we have
Moreover, we have just proved that , and so . Accordingly, since , we have . Hence, since , we get . Consequently, the condition (C2) of Lemma 2.2 is clear. As consequence, , as desired.

*Remark 2.3. *From the proof of Theorem 2.1, we can easily see that
provided is a semiartinian ring.

Corollary 2.4. *Let be a semiartinian ring with finite Gorenstein global dimension. Then,
**
where ranges over all simple -modules.*

*Proof. *It is sufficient to prove the inequality , where ranges over all simple -modules. We may suppose finite. By Remark 2.3, . Using [8, Theoremββ2.20], for any -module . Consequently, , as desired.

Recall that ring is called *quasi-Frobenius* if it is self injective and artinian.

Corollary 2.5. *The following conditions are equivalent: *(1)* is quasi-Frobenius, *(2)* is a semiartinian ring and every simple -module is Gorenstein projective.*

*Proof. *(1)*β*(2) Follows immediately from [5, Propositionββ2.10] and Corollary 2.4.

(2)*β*(1) From Remark 2.3, every projective module is injective. Thus, by [14, Theoremββ7.55], is quasi-Frobenius.

Since every perfect ring is semiartinian by [10, Propositionββ5.1],we have the following corollary.

Corollary 2.6. *If is a perfect ring with Jacobson radical , then
**
Moreover, if is not quasi-Frobenius ring then, .*

*Proof. *Using [8, Propositionββ2.19] and its dual version, this result is immediate since every simple -module is a direct summand of the -module by [15, Theoremββ9.3.4].

The second main result of this paper computes the weak Gorenstein global dimension of semiartinian rings via the Gorenstein flat dimensions of simple modules.

Theorem 2.7. *Let be a semiartinian ring and let be a positive integer. The following conditions are equivalent: *(1)*, *(2)* for all simple -modules , *(3)* for all simple -modules and all injective -modules .**
Consequently, , where ranges ranges over all simple -modules.*

We need the following lemmas.

Lemma 2.8. *Let be an -module. If then for all and every injective -modules .*

*Proof. *Using the definition of Gorenstein flat module, the case is clear. For , we consider an -step flat resolution of , and we use the start case.

Lemma 2.9 (Theoremββ2.4, [11]). *Let be a ring and a positive integer. The following conditions are equivalent: *(1)*, *(2)* for every injective module .*

*Proof of Theorem 2.7. *(1)*β*(2) Follows immediately from the definition of weak Gorenstein global dimension.

(2)*β*(3) Follows from Lemma 2.8.

(3)*β*(1) By [12, Lemmaββ4.2(1)], for every injective -module since for all simple -modules . Hence, this implication follows from Lemma 2.9.

Recall that a ring is called -*ring* if every injective module is flat; see [16].

Corollary 2.10. *Let be a semiartinian ring. The following are equivalent: *(1)* is an -ring. *(2)*Every simple -module is Gorenstein flat. *(3)* is coherent and is flat for every simple -module where is the injective envelope of .*

*Proof. *Using Lemma 2.8, a ring is -ring if and only if . Hence, the equivalence (1)(2) is an immediate consequence of Theorem 2.7.

(3)*β*(1) Follows from [16, Theoremββ3.8].

(1)*β*(3) From Lemma 2.9, if is an -ring, then every -module is Gorenstein flat. Then, by [17, Theoremββ6], is coherent and self -injective (i.e., for every finitely presented -module ). Thus, by [16, Theoremββ3.8], is flat for every simple -module , where is the injective envelope of , and certainly is coherent.

*Remark 2.11. *Note that the equivalence of and in the above corollary does not need that be semiartinian, see [16, Theoremββ3.8] and [18, Propositionββ4.2].

Lemma 2.12. *Let be an exact sequence of modules over a coherent ring . Then, with equality if .*

*Proof. *Using [8, Theoremββ3.15] and [8, Theoremββ3.14], the proof is similar to that of [19, Corollaryββ2, page 135].

Corollary 2.13. *If is a perfect coherent ring with Jacobson radical , then . Moreover, if is not -ring then, .*

*Proof. *Using [8, Propositionββ3.13], the first statement of this result is an immediate consequence of Theorem 2.7 since every simple -module is a direct summand of the -module by [15, Theoremββ9.3.4].

Suppose that is not -ring. Then, by Lemma 2.9, . Therefore, by Lemma 2.12, we deduce from the exact sequence
that .

#### Acknowledgment

The authors would like to express their sincere thanks to the referees for their helpful suggestions and comments.

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