The Weak (Gorenstein) Global Dimension of Coherent Rings with Finite Small Finitistic Projective Dimension

Alhazmy, Khaled; Ahmed Almahdi, Fuad Ali; El Haddaoui, Younes; Mahdou, Najib

doi:https://doi.org/10.1155/2024/4896819

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Volume 2024 | Article ID 4896819 | https://doi.org/10.1155/2024/4896819

The Weak (Gorenstein) Global Dimension of Coherent Rings with Finite Small Finitistic Projective Dimension

Khaled Alhazmy,¹Fuad Ali Ahmed Almahdi,¹Younes El Haddaoui,²and Najib Mahdou²

Academic Editor: R. Sundareswaran

Received04 Jan 2024

Revised26 Mar 2024

Accepted27 Mar 2024

Published17 Apr 2024

Abstract

The small finitistic dimension of a ring is determined as the supremum projective dimensions among modules with finite projective resolutions. This paper seeks to establish that, for a coherent ring with a finite weak (resp. Gorenstein) global dimension, the small finitistic dimension of is equal to its weak (resp. Gorenstein) global dimension. Consequently, we conclude some new characterizations for (Gorenstein) von Neumann and semihereditary rings.

1. Introduction

In this paper, we assume all rings are commutative with identity, and all modules are unitary. Let be a ring, and an -module. As usual, we use and to represent the classical projective dimension and flat dimension of , respectively. The weak dimension of is defined as , and denotes the total quotient ring of .

The -dimension was initially introduced, by Auslander and Bridger [1], for commutative Noetherian rings. This concept was subsequently expanded to modules over any ring by Enochs and Jenda [2, 3] through the introduction of Gorenstein projective, injective, and flat modules. The investigation of homological dimensions based on these modules was pursued in [4].

Let us consider a ring . A module is termed Gorenstein projective, for short -projective, if there exists an exact sequence of projective modulessuch that is isomorphic to the image of the map , and the functor maintains the exactness of whenever is a projective module. This sequence is termed a complete projective resolution.

Similarly, a module is is termed Gorenstein flat, for short -flat, if there exists an exact sequence of flat modules:such that is isomorphic to the image of the map , and the functor preserves the exactness of whenever is an injective module. The sequence is called a complete flat resolution.

Gorenstein projective and flat dimensions, denoted by and , respectively, are defined based on resolutions ([4, 5]).

The weak Gorenstein global dimension of a ring is defined as follows:

It is important to observe that for a given ring , the weak Gorenstein global dimension is bounded above by the weak dimension , and the two coincide if is finite.

Let be a ring and be a module. An exact sequencewhere is finitely generated projective modules, is called a finite projective resolution ( for short) of .

The small finitistic dimension of a ring , denoted , is defined to be the supremum of projective dimensions of modules with . In the case of a Noetherian local ring , Auslander and Buchweitz in [6] showed that coincides with the depth of . It is evident that if and only if any module with an is projective. Equivalently, this condition holds if and only if is projective whenever there exists an exact sequence , where and are finitely generated projective modules.

In the context of a coherent ring , the small finitistic projective dimension assumes a more tractable form, namely,

Similarly, for the weak global dimension of a coherent ring , a nice description is given by

Similarly, we define the small finitistic Gorenstein projective dimension of a ring , as follows:

The close relation between the small finitistic projective dimension, the weak global dimension, and the weak Gorenstein global dimension renders it natural to track the possible values of and for a given value of .

The aim of this paper is to answer the following question:

Question. For a coherent ring with , what values can the weak (resp. Gorenstein) global dimension take?

We begin by establishing the equality of and for a coherent ring (in Theorem 1). This leads to new characterizations of von Neumann regular and semihereditary rings. A particular focus is on rings with zero . It has been demonstrated that when is Noetherian with zero Krull dimension, (in ([6], Theorem 1.6)). Interestingly, this result extends beyond the Noetherian assumption, as proved in [7], Proposition 3.14. However, Example 2 illustrates that the converse implication is not valid. Theorem 10 establishes the equality of and , both bounded by . In addition, in the case of a coherent ring , these three dimensions coincide. Consequently, we found new characterizations for rings with small . Finally, Proposition 17 presents a new characterization of quasi-Frobenius rings through the utilization of Nagata rings.

2. The Weak (Gorenstein) Global Dimension of Coherent Rings with Finite Small Finitistic Projective Dimension

Generally, for a ring , , with equality when is local, coherent, and regular, as shown in [8], Lemma 3.1. A ring is said to be regular if every finitely generated ideal of has finite projective dimension, as defined in [8]. This concept has been extensively explored in the context of coherent rings. Notably, coherent rings having finite weak global dimension are regular. Nevertheless, it is essential to note that there exist coherent rings, including local ones, possessing an infinite weak global dimension while maintaining regularity.

The first main result of this paper drops the “local” condition in Glaz’s result [8], Lemma 3.1.

Theorem 1. Let be a regular coherent ring. Then, .

Proof. Consider a coherent regular ring . It is evident that . Now, let’s set . Consider a finitely generated ideal of . Then, since is regular, . As is coherentConsequently, . Hence, for any module . By ([9], Theorems 2.6.1 and 2.6.3), . Therefore, . Consequently, we conclude that , as desired.
Let be a finitely generated ideal of a ring . If , then is called semiregular. When is the only finitely generated semiregular ideal of , then is called a ring. It is proven, in ([10], Proposition 2.2), that a ring is a ring if and only if is zero. Hence, fPD mesures how far a ring to be .
In [9], Glaz introduced the concept of -rings. A ring is a -ring (or has the property (P)) if for each finitely generated proper ideal of . Glaz pioneered the exploration of the homological properties of local -rings and demonstrated that a local ring is a -ring if and only if fPD . The aforementioned result has been further generalized in ([11], Theorem 1) to apply to arbitrary rings (not necessarily local).
We conclude the following corollaries.

Corollary 2. If is a ring, then the following are equivalent:(1) is a von Neumann regular ring (i.e., ).(2) is coherent, , and .(3) is a coherent -ring and.(4) is coherent regular with .(5) is a coherent regular -ring.(6) is a coherent regular ring.

Corollary 3. If is a ring, then the following are equivalent:(1) is a semihereditary ring (i.e., is coherent and ).(2) is coherent, , and .(3) is coherent regular with .

Remark 4. It is established in ([12], Corollary 3.2) that, for a ring , if then finitey generated flat modules are projective. However, this assertion does not hold in general. Take, for instance, a von Neumann regular ring which is not semisimple. Since is not Noetherian, has a nonfinitely generated ideal . The module is finitely generated flat that is not projective since it is not of finitely presented.

Auslander and Buchsbaum, in ([6], Theorem 1.6), established that for a Noetherian ring , is less than or equal to the Krull dimension of , denoted by . Consequently, when , it implies that . However, this conclusion holds true even in cases where is not necessarily Noetherian as shown by Wang, Zhou, and Chen in ([7], Proposition 3.14).

In ([13], Problem 1b), Cahen et al. asked if is always zero for a total ring of quotients . Rings with zero Krull dimension constitute a subclass of total rings of quotients where is indeed zero. However, a recent study in [7] provided a negative answer to this question.

Note that a ring with does not need to be coherent or have finite .

Example 1. (1)Let be a nonsemisimple quasi-Frobenius ring. Then, is Noetherian, , and .(2)Let be a noncoherent ring with zero Krull dimension (see, for instance, ([14], Example 2.8),). Then, by ([7], Proposition 3.14), .(3)Set . Let be a proper finitely generated ideal of generated by . Set and . Clearly, . Since is proper, or . Suppose, for example, that . Then, since is a -ring, there exists a nonzero element such that . Hence, . So, is a -ring, and then . However, is not coherent since is not, and .Recall that a ring is called McCoy (or satisfies Property ) if for each finitely generated ideal consisting of zero divisors of . McCoy rings include Noetherian rings, rings with Krull dimension zero, and graded rings (in particular, polynomial rings). Next, we give a new characterization of McCoy rings.

Proposition 5. A ring is McCoy if and only if . In particular, if is a total ring of quotients then is McCoy if and only if .

Proof. By ([15], Corollary 2.6), is McCoy if and only if is McCoy.
Clearly, if , then is a -ring, and, in particular, it is McCoy. Consequently, is McCoy. Conversely, if is McCoy, then so is . Since is a total ring of quotients, every proper ideal of consists only of zero divisors. Hence, is a -ring, as desired.
A ring with , even Noetherian, does not necessarily have a zero Krull dimension, as shown by the next example.

Example 2. Let be a field. Consider the additive groupequipped with multiplication defined asThis forms a commutative ring with unity , known as the trivial extension of by the -module . According to ([16], Theorems 3.2 and 4.8), , and is a local Noetherian ring since is local Noetherian, and is a finitely generated -module. Moreover, the maximal ideal of is . For each , we have . Consequently, every nonunit element of is a zero divisor, establishing as a total ring of quotients. According to ([10], Proposition 2.3), is a ring, and thus, .

Let be a ring, and consider an ideal of . In accordance with [17], is designated as a -ideal if it is finitely generated, and the natural homomorphism is an isomorphism. Let denotes the set of -ideals of . Consider a module and set

It is evident that forms a submodule of . A module is said to be -torsion-free (resp. -torsion) if (resp. ). A -torsion-free module is said to be a -module if for any . When every ideal of is a -ideal, we say that is .

The notion of rings is related to rings with small finitistic projective dimension . Let be a ring. Wang et al. in ([10], Proposition 2.2 and Theorem 3.2) proved that is equivalent to being a ring and (where is the ring of finite fractions of ). It is also proved, in ([18], Corollary 3.7), that is a ring if and only if . Hence, we can rewrite Corollary 3 as follows:

Proposition 6. If is a ring, then the following are equivalent:(1) is a semi-hereditary ring.(2) is a coherent ring with .(3) is a coherent regular ring.

In particular, is a Prüfer domain (i.e., a semihereditary domain) if and only if is a coherent regular domain.

In the previous result, the particular case is exactly ([19], Proposition 3.1 ). Recall that a ring is called a Prüfer ring if every finitely generated regular ideal is invertible. Over a domain, the two definitions of Prüfer domains coincide. It is also well known that semihereditary rings are Prüfer rings. Hence, coherent regular rings are Prüfer rings. However, as mentioned in [19], Prüfer rings need not be regular. For example, is a local Noetherian Prüfer ring with infinite (weak) global dimension, and so it is not a regular ring.

Using Corollary 3 and Proposition 6, we conclude the following corollary:

Corollary 7. If is a domain, then the following are equivalent:(1) is a Dedekind domain.(2) is Noetherian, , and has a finite global dimension.(3) is Noetherian regular domain.

Remark 8. Recall the classical definition of regularity for Noetherian rings: A local Noetherian ring is regular if it has a finite global dimension. A general Noetherian ring is regular if it is locally regular. It’s important to note that for a Noetherian ring, the two definitions of regularity, the one provided in [8] and the classical one, coincide. Therefore, Corollary 7 is partially ([20], Proposition 3.6).

Now, we define a Gorenstein analogue for the .

Definition 9. Let be a ring and be a module. The small finitistic Gorenstein projective dimension of a ring , denoted , is defined as follows:The next result compares fPD(−) with fGPD(−), and wGdim(−).

Theorem 10. Let be a ring. Then,(1).(2)If is coherent with , then .

The following lemmas are required.

Lemma 11. Let be a finitely generated -projective module. There is a short exact sequence , where is a finitely generated projective module and is a finitely generated -projective module.

Proof. This is exactly ([21], Lemma 2.9) with the precision that in the proof can be taken to be finitely generated.

Lemma 12. Let be a ring and be a module with finite -projective dimension . If has a , then there exists an epimorphism , where is a -projective module with , and is module with and .

Proof. Since has a , we can consider an exact sequencewhere all is finitely generated projective modules and is a module with . According to ([4], Theorem 2.20), is -projective. Using Lemma 11, we obtain an exact sequence:where all is finitely generated projective and is a -projective module with , and such that the functor maintains the exactness of this sequence when is projective.
This enables the construction of homomorphisms for and such that the following diagram is commutative.By ([22], Proposition 1.4.14), we get an exact sequence.resulting in the exactness of the sequenceIt is worth noting that has a . Consequently, the kernel of satisfies and has a (by ([8], Theorem 2.1.2)).

Proof of Theorem 13. (1)We claim the inequality . Let us assume . Since every -module has a is finitely presented with finite projective dimension, by ([4], Proposition 2.27), we have . Now, we aim to establish . Let us assume . Consider a module with a and finite -projective dimension. Since every -module has a is infinitely presented, by ([23], Theorem 3.3), we have . This implies , as desired. We claim . We may assume . Consider a module with s and finite -projective dimension. According to Lemma 12, there exists an exact sequence , where is -projective and is a module with and finite projective dimension. Hence, . By Lemma 11, there exists a short exact sequence , where is finitely generated projective and is finitely generated -projective. The pushout diagram with exact rows and columns is given by Clearly, has a and a finite projective dimension. Thus, . Using the short exact sequence and ([4], Theorem 2.22), for each integer , we obtain for all projective modules . Once again, by ([4], Theorem 2.22), . Consequently, .(2)Let be a positive integer. Recall that a ring is said to be -FC if is coherent and (the -injective dimension of ). Suppose that a ring is coherent with . Using ([24], Theorem 3.8) and ([20], Theorem 10), we get thatThus, is a -FC ring. In accordance with ([25], Theorem 7), we conclude that .

Corollary 14. If is a ring. Then, if and only if every finitely generated projective submodule of a -projective module is a direct summand.

Proof. This follows from ([21], Lemma 2.8) and the fact that if and only if is a -ring.
Recall that a ring is called G-von Neumann regular (resp. G-semi-hereditary) if (resp. is coherent and ).

Corollary 15. If is a ring, then the following are equivalent:(1) is a G-semi-hereditary ring.(2) is coherent, , and .

Let denote the set of finitely generated semiregular ideals of a ring . The concept of the ring of finite fractions for a ring , denoted as , was explored, by Lucas in [26]:

The inclusions were established, and in the case where is an integral domain, it was shown that serves as the quotient field of . Moreover, in [27], it was demonstrated that if and only if every finitely generated semiregular ideal of is a -ideal. For further details, please refer to [10, 26, 27].

Theorem 16. If is a ring, then the following are equivalent:(1) is Gorenstein von Neuman regular.(2) is coherent, , and .(3) is a coherent -ring and.(4) is a coherent ring with .(5) is G-semi-hereditary and .(6) is a G-semi-hereditary -ring.(7) is a G-semi-hereditary ring.(8) is a G-semi-hereditary ring and .

Proof. The equivalence between (1), (2), (3), (4), (5), (6), and (7) follows immediately from Theorem 10 and the fact that if and only if is if and only if is a -ring.
This follows from ([10], Proposition 2.2).
Suppose is a G-semi-hereditary ring such that . Then, by Theorem 10, . Using ([10], Corollary 3.3), is a ring.
Consider a commutative ring and a polynomial . The content of , denoted by , refers to the ideal of generated by the coefficients of .
Set , the set of polynomials with unit content. The Nagata ring is obtained by localizing the polynomial ring with respect to ; that is .

Proposition 17. If is a ring, then the following are equivalent:(1) is quasi-Frobenius.(2) is a Noetherian G-semi-hereditary ring and .

Proof. Note that is quasi-Frobenius if and only if is a Noetherian G-von Neumann regular (by ([28], Theorem 2.2) and ([21], Corollary 2.11)). Moreover, if is Noetherian (and thus a McCoy ring), then, by ([26], Theorem 3.2), is equivalent to . Hence, the desired conclusion follows from Theorem 16.

Data Availability

The study of this article does not require any data.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through Small group Research Project under grant no. RGP1/276/44. The authors extend also their sincere gratitude to the referees for their invaluable suggestions and comments, which have greatly improved this paper.

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Copyright © 2024 Khaled Alhazmy et al. 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.

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