# (Weak) Gorenstein Global Dimension of Semiartinian Rings

Authors: Mohammed Tamekkante; Mohamed Chhiti
Journal: ISRN Algebra
 (2011)

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## Abstract

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


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## Body

## 1. Introduction

Throughout the paper, all rings are commutative with identity. LetR 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 of M.For a two-sided Noetherian ringR, 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-modules M, 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 anR-module M is called Gorenstein projective, if there exists an exact sequence of projective R-modules (1.1)P:⋯⟶P1⟶P0⟶P0⟶P1⟶⋯such thatM≅Im(P0→P0) and such that the functor HomR(-,Q) leaves P exact whenever Q is a projective R-module. The complex P is called a complete projective resolution.The Gorenstein injectiveR-modules are defined dually.AnR-module M is called Gorenstein flat, if there exists an exact sequence of flat R-modules: (1.2)F:⋯⟶F1⟶F0⟶F0⟶F1⟶⋯such thatM≅Im(F0→F0) and such that the functor I⊗R- leaves F exact whenever I is a right injective R-module. The complex F is called a complete flat resolution.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 (1.3)sup⁡{GpdR(M)∣Mis a(left)R-module}=sup⁡{GidR(M)∣Mis a(left)R-module}.
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 (1.4)l.wGgldim(R)=sup⁡{GfdR(M)∣Mis a(left)R-module},
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 anR-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).


## 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.
LetR be a semiartinian ring and  n a positive integer. The following conditions are equivalent:   (1)
Ggldim(R)≤n,(2)
supC{GpdR(C),GidR(C)}≤n where C ranges ranges over all simple R-modules,(3)
ExtRn+1(C,P)=TorRn+1(I,C)=0 for all simple R-modules C, all projective R-modules P, and all injective R-modules I. Consequently,
(2.1)Ggldim(R)=sup⁡C⁡{GpdR(C),GidR(C)},
where C ranges ranges over all simple R-modules.To prove this theorem, we need the following lemma.Lemma 2.2 (Theorem2.1, [11]).
LetR 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 Theorem2.1.
(1)⇒(2) Clear by the definition of Ggldim(R).
(2)⇒(3) By [8, Theorem  2.20], ExtRn+1(C,P)=0 for all i>n and all simple R-module C and all projective module P since GpdR(C)≤n. Let I be an injective R-module. By [8, Theorem  2.22], ExtRn+1(I,C)=0 for all i>n and all simple R-module C since GidR(C)≤n. Then, by [12, Lemma  3.1(1)], TorRn+1(I,C)=0 for every simple R-module C.
(3)⇒(1) Let P be a projective R-module. By [12, Lemma  4.2(2)], idR(P)≤n since ExtRn+1(C,P)=0 for all simple R-modules C. 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)], fdR(I)≤n since TorRn+1(I,C)=0 for all simple R-module C. On the other hand, from [13, Theorem  7.2.5(2) and Corollary  7.2.6(1)⇔(2)], we have
(2.2)(a)∶=sup⁡{pdR(F)∣Fis a flatR-module}≤(b)∶=sup⁡{pdR(M)∣Mis anR-module withpdR(M)<∞}≤(c)∶=sup⁡{idR(P)∣Pis a projectiveR-module}.
Moreover, we have just proved that (c)≤n, and so (a)≤n. Accordingly, since fdR(I)≤n, we have pdR(I)<∞. Hence, since (b)≤n, we get pdR(I)≤n. Consequently, the condition (C2) of Lemma 2.2 is clear. As consequence, Ggldim(R)≤n, as desired.Remark 2.3.
From the proof of Theorem2.1, we can easily see that
(2.3)sup⁡{idR(P)∣Pis a projectiveR-module}≤sup⁡{GpdR(C)∣Cis a simpleR-module},
provided R is a semiartinian ring.Corollary 2.4.
LetR be a semiartinian ring with finite Gorenstein global dimension. Then,
(2.4)Ggldim(R)=sup⁡C{GpdR(C)},
where C ranges over all simple R-modules.Proof.
It is sufficient to prove the inequalityGgldim(R)≤supC{GpdR(C)}, where C ranges over all simple R-modules. We may suppose n=supC{GpdR(C)} finite. By Remark 2.3, sup{idR(P)∣Pis a projectiveR-module}≤n. Using [8, Theorem  2.20], GpdR(M)≤n for any R-module M. Consequently, Ggldim(R)≤n, as desired.Recall that ringR is called quasi-Frobenius if it is self injective and artinian.Corollary 2.5.
The following conditions are equivalent:(1)
R is quasi-Frobenius,(2)
R is a semiartinian ring and every simple R-module is Gorenstein projective.Proof.
(1)⇒(2) Follows immediately from [5, Proposition  2.10] and Corollary 2.4.
(2)⇒(1) From Remark 2.3, every projective module is injective. Thus, by [14, Theorem  7.55], R is quasi-Frobenius.Since every perfect ring is semiartinian by [10, Proposition  5.1],we have the following corollary.Corollary 2.6.
IfR is a perfect ring with Jacobson radical  J, then
(2.5)Ggldim(R)=sup⁡{GpdR(RJ),GidR(RJ)}.
Moreover, if R is not quasi-Frobenius ring then, Ggldim(R)=1+sup{GpdR(J),GidR(J)}.Proof.
Using [8, Proposition  2.19] and its dual version, this result is immediate since every simple R-module is a direct summand of the R-module R/J by [15, Theorem  9.3.4].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.
LetR be a semiartinian ring and let  n be a positive integer. The following conditions are equivalent:   (1)
wGgldim(R)≤n,(2)
GfdR(C)≤n for all simple R-modules C,(3)
TorRn+1(I,C)=0 for all simple R-modules C and all injective R-modules I.
Consequently,wGgldim(R)=supC{GfdR(C)}, where C ranges ranges over all simple R-modules.We need the following lemmas.Lemma 2.8.
LetM be an  R-module. If  GfdR(M)≤n then  TorRi(I,M)=0 for all  i>n and every injective  R-modules  I.Proof.
Using the definition of Gorenstein flat module, the casen=0 is clear. For n>0, we consider an n-step flat resolution of M, and we use the start case.Lemma 2.9 (Theorem  2.4, [11]).
LetR be a ring and  n a positive integer. The following conditions are equivalent: (1)
wGgldim(R)≤n,(2)
fdR(I)≤n for every injective module I.Proof of Theorem2.7.
(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)], fdR(I)≤n for every injective R-module I since TorRn+1(I,C)=0 for all simple R-modules C. Hence, this implication follows from Lemma 2.9.Recall that a ring is calledIF-ring if every injective module is flat; see [16].Corollary 2.10.
LetR be a semiartinian ring. The following are equivalent:   (1)
R is an IF-ring.(2)
Every simpleR-module is Gorenstein flat.(3)
R is coherent and E(C) is flat for every simple R-module C where E(C) is the injective envelope of C.Proof.
Using Lemma2.8, a ring R is IF-ring if and only if wGgldim(R)=0. 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 R is an IF-ring, then every R-module is Gorenstein flat. Then, by [17, Theorem  6], R is coherent and self FP-injective (i.e., ExtR1(M,R)=0 for every finitely presented R-module M). Thus, by [16, Theorem  3.8], E(C) is flat for every simple R-module C, where E(C) is the injective envelope of C, and certainly R is coherent.Remark 2.11.
Note that the equivalence of(1) and (3) in the above corollary does not need that R be semiartinian, see [16, Theorem  3.8] and [18, Proposition  4.2].Lemma 2.12.
Let0→N→N′→N′′→0 be an exact sequence of modules over a coherent ring  R. Then,  GpdR(N′′)≤max{GpdR(N′),GpdR(N)+1} with equality if  GpdR(N′)≠GpdR(N).Proof.
Using [8, Theorem  3.15] and [8, Theorem  3.14], the proof is similar to that of [19, Corollary  2, page 135].Corollary 2.13.
IfR is a perfect coherent ring with Jacobson radical  J, then  wGgldim(R)=GfdR(R/J). Moreover, if  R is not  IF-ring then,  wGgldim(R)=1+sup{GfdR(J)}.Proof.
Using [8, Proposition  3.13], the first statement of this result is an immediate consequence of Theorem 2.7 since every simple R-module is a direct summand of the R-module R/J by [15, Theorem  9.3.4].
Suppose thatR is not IF-ring. Then, by Lemma 2.9, wGgldim(R)=GfdR(R/N)>0. Therefore, by Lemma 2.12, we deduce from the exact sequence
(2.6)0⟶N⟶R⟶RN⟶0
that wGgldim(R)=GpdR(R/N)=1+GpdR(N).



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*Source: 497831-2011-06-27.xml*