Research Article | Open Access
Mohammed Tamekkante, Mohamed Chhiti, "(Weak) Gorenstein Global Dimension of Semiartinian Rings", International Scholarly Research Notices, vol. 2011, Article ID 497831, 6 pages, 2011. https://doi.org/10.5402/2011/497831
(Weak) Gorenstein Global Dimension of Semiartinian Rings
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.
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  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.  introduced the Gorenstein flat dimension. Some references are [2–8].
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.
In , 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 .
In , 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, ). 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)⇒(2) Clear by the definition of .
(2)⇒(3) By [8, Theorem 2.20], for all and all simple -module and all projective module since . Let be an injective -module. By [8, Theorem 2.22], for all and all simple -module since . Then, by [12, Lemma 3.1(1)], for every simple -module .
(3)⇒(1) Let be a projective -module. By [12, Lemma 4.2(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, Lemma 4.2(1)], since for all simple -module . On the other hand, from [13, Theorem 7.2.5(2) and Corollary 7.2.6(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.
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.
Since every perfect ring is semiartinian by [10, Proposition 5.1],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, .
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 (Theorem 2.4, ). 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)⇒(2) Follows immediately from the definition of weak Gorenstein global dimension.
(2)⇒(3) Follows from Lemma 2.8.
(3)⇒(1) By [12, Lemma 4.2(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 .
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)⇒(1) Follows from [16, Theorem 3.8].
(1)⇒(3) From Lemma 2.9, if is an -ring, then every -module is Gorenstein flat. Then, by [17, Theorem 6], is coherent and self -injective (i.e., for every finitely presented -module ). Thus, by [16, Theorem 3.8], is flat for every simple -module , where is the injective envelope of , and certainly is coherent.
Lemma 2.12. Let be an exact sequence of modules over a coherent ring . Then, with equality if .
Corollary 2.13. If is a perfect coherent ring with Jacobson radical , then . Moreover, if is not -ring then, .
Proof. Using [8, Proposition 3.13], 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, Theorem 9.3.4].
Suppose that is not -ring. Then, by Lemma 2.9, . Therefore, by Lemma 2.12, we deduce from the exact sequence that .
The authors would like to express their sincere thanks to the referees for their helpful suggestions and comments.
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