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

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


Introduction
In this paper, we assume all rings are commutative with identity, and all modules are unitary.Let R be a ring, and M an R-module.As usual, we use pd R (M) and fd R (M) to represent the classical projective dimension and fat dimension of M, respectively.Te weak dimension of R is defned as wdim(R) � sup fd R (M) | M is an R − module  , and T(R) denotes the total quotient ring of R.
Te G-dimension was initially introduced, by Auslander and Bridger [1], for commutative Noetherian rings.Tis concept was subsequently expanded to modules over any ring by Enochs and Jenda [2,3] through the introduction of Gorenstein projective, injective, and fat modules.Te investigation of homological dimensions based on these modules was pursued in [4].
Let us consider a ring R. A module M is termed Gorenstein projective, for short G-projective, if there exists an exact sequence of projective modules such that M is isomorphic to the image of the map Q 0 ⟶ Q 0 , and the functor Hom R (− , Q) maintains the exactness of Q whenever Q is a projective module.Tis sequence Q is termed a complete projective resolution.
Similarly, a module M is is termed Gorenstein fat, for short G-fat, if there exists an exact sequence of fat modules: such that M is isomorphic to the image of the map F 0 ⟶ F 0 , and the functor I⊗ R − preserves the exactness of F whenever I is an injective module.Te sequence F is called a complete fat resolution.
Te weak Gorenstein global dimension of a ring R is defned as follows: It is important to observe that for a given ring R, the weak Gorenstein global dimension wGdim(R) is bounded above by the weak dimension wdim(R), and the two coincide if wdim(R) is fnite.
Let R be a ring and M be a module.An exact sequence where P i is fnitely generated projective modules, is called a fnite projective resolution (fpr for short) of M.
Te small fnitistic dimension of a ring R, denoted fPD(R), is defned to be the supremum of projective dimensions of modules with fpr.In the case of a Noetherian local ring R, Auslander and Buchweitz in [6] showed that fPD(R) coincides with the depth of R. It is evident that fPD(R) � 0 if and only if any module M with an fpr is projective.Equivalently, this condition holds if and only if M is projective whenever there exists an exact sequence 0 ⟶ Q 1 ⟶ Q 0 ⟶ M ⟶ 0, where Q 0 and Q 1 are fnitely generated projective modules.
In the context of a coherent ring R, the small fnitistic projective dimension fPD(R) assumes a more tractable form, namely, Similarly, for the weak global dimension of a coherent ring R, a nice description is given by Similarly, we defne the small fnitistic Gorenstein projective dimension of a ring R, as follows: Te close relation between the small fnitistic projective dimension, the weak global dimension, and the weak Gorenstein global dimension renders it natural to track the possible values of wdim(R) and wGdim(R) for a given value of fPD(R).
Te aim of this paper is to answer the following question: Question.For a coherent ring with fPD(R) � n, what values can the weak (resp.Gorenstein) global dimension R take?
We begin by establishing the equality of fPD(R) and wdim(R) for a coherent ring R (in Teorem 1).Tis leads to new characterizations of von Neumann regular and semihereditary rings.A particular focus is on rings with zero fPD.It has been demonstrated that when R is Noetherian with zero Krull dimension, fPD(R) � 0 (in ( [6], Teorem 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.Teorem 10 establishes the equality of fPD(R) and fGPD(R), both bounded by wGdim(R).In addition, in the case of a coherent ring R, these three dimensions coincide.Consequently, we found new characterizations for rings with small wGdim.Finally, Proposition 17 presents a new characterization of quasi-Frobenius rings through the utilization of Nagata rings.

The Weak (Gorenstein) Global Dimension of Coherent Rings with Finite Small Finitistic Projective Dimension
Generally, for a ring R, fPD(R) ≤ wdim(R), with equality when R is local, coherent, and regular, as shown in [8], Lemma 3.1.A ring is said to be regular if every fnitely generated ideal of R has fnite projective dimension, as defned in [8].Tis concept has been extensively explored in the context of coherent rings.Notably, coherent rings having fnite weak global dimension are regular.Nevertheless, it is essential to note that there exist coherent rings, including local ones, possessing an infnite weak global dimension while maintaining regularity.Te frst main result of this paper drops the "local" condition in Glaz's result [8], Lemma 3.1.
Let J be a fnitely generated ideal of a ring R. If Hom R (R/J, R) � 0, then J is called semiregular.When R is the only fnitely generated semiregular ideal of R, then R is called a DQ ring.It is proven, in ([10], Proposition 2.2), that a ring R is a DQ ring if and only if fPD(R) is zero.Hence, fPD mesures how far a ring to be DQ.
In [9], Glaz introduced the concept of P-rings.A ring R is a P-ring (or has the property (P)) if ann R (I) ≠ (0) for each fnitely generated proper ideal I of R. Glaz pioneered the exploration of the homological properties of local P-rings and demonstrated that a local ring R is a P-ring if and only if fPD (R) � 0. Te aforementioned result has been further generalized in ([11], Teorem 1) to apply to arbitrary rings (not necessarily local).
We conclude the following corollaries.
(5) R is a coherent regular P-ring.(6) R is a coherent regular DQ ring.

Corollary 3.
If R is a ring, then the following are equivalent: (1) R is a semihereditary ring (i.e., R is coherent and wdim Remark 4. It is established in ([12], Corollary 3.2) that, for a ring R, if fPD(R) � 0 then fnitey generated fat modules are projective.However, this assertion does not hold in general.Take, for instance, a von Neumann regular ring R which is not semisimple.Since R is not Noetherian, R has a nonfnitely generated ideal I. Te module R/I is fnitely generated fat that is not projective since it is not of fnitely presented.
Auslander and Buchsbaum, in ([6], Teorem 1.6), established that for a Noetherian ring R, fPD(R) is less than or equal to the Krull dimension of R, denoted by dim(R).Consequently, when dim(R) � 0, it implies that fPD(R) � 0. However, this conclusion holds true even in cases where R 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 fPD(R) is always zero for a total ring of quotients R. Rings with zero Krull dimension constitute a subclass of total rings of quotients where fPD is indeed zero.However, a recent study in [7] provided a negative answer to this question.
Note that a ring R with fPD(R) � 0 does not need to be coherent or have fnite wdim(R).

Recall that a ring
equipped with multiplication defned as Journal of Mathematics
Let R be a ring, and consider an ideal I of R. In accordance with [17], I is designated as a GV-ideal if it is fnitely generated, and the natural homomorphism R ⟶ Hom R (I, R) is an isomorphism.Let GV(R) denotes the set of GV-ideals of R. Consider a module M and set It is evident that tor GV (M) forms a submodule of M. A module M is said to be GV-torsion-free Te notion of DW rings is related to rings with small fnitistic projective dimension ≤ 1.Let R be a ring.Wang et al. in ([10], Proposition 2.2 and Teorem 3.2) proved that fPD(R) � 0 is equivalent to R being a DW ring and R � Q 0 (R) (where Q 0 (R) is the ring of fnite fractions of R ).It is also proved, in ( [18], Corollary 3.7), that R is a DW ring if and only if fPD(R) ≤ 1.Hence, we can rewrite Corollary 3 as follows: Proposition 6.If R is a ring, then the following are equivalent: (1) R is a semi-hereditary ring.
(3) R is a coherent regular DW ring.
In particular, R is a Prüfer domain (i.e., a semihereditary domain) if and only if R is a coherent regular DW domain.
In the previous result, the particular case is exactly ([19], Proposition 3.1 (2) ⟺ (6)).Recall that a ring R is called a Prüfer ring if every fnitely generated regular ideal is invertible.Over a domain, the two defnitions of Prüfer domains coincide.It is also well known that semihereditary rings are Prüfer rings.Hence, coherent regular DW rings are Prüfer rings.However, as mentioned in [19], Prüfer rings need not be regular.For example, Z/4Z is a local Noetherian Prüfer ring with infnite (weak) global dimension, and so it is not a regular ring.
Using Corollary 3 and Proposition 6, we conclude the following corollary: A general Noetherian ring R is regular if it is locally regular.It's important to note that for a Noetherian ring, the two defnitions of regularity, the one provided in [8] and the classical one, coincide.Terefore, Corollary 7 is partially ( [20], Proposition 3.6).Now, we defne a Gorenstein analogue for the fPD(− − ).Defnition 9. Let R be a ring and M be a module.Te small fnitistic Gorenstein projective dimension of a ring R, denoted fGPD(R), is defned as follows: Te next result compares fPD(− ) with fGPD(− ), and wGdim(− ).
Theorem 10.Let R be a ring.Ten, (1) Te following lemmas are required.
Lemma 11.Let X be a fnitely generated G-projective module.Tere is a short exact sequence 0 ⟶ X ⟶ P ⟶ X ′ ⟶ 0, where P is a fnitely generated projective module and X ′ is a fnitely generated G-projective module.
Proof.Tis is exactly ([21], Lemma 2.9) with the precision that in the proof X ′ can be taken to be fnitely generated.□ Lemma 12. Let R be a ring and M be a module with fnite G-projective dimension n ≥ 1.If M has a fpr, then there exists an epimorphism ϵ: G 0 ↠M, where G 0 is a G-projective module with fpr, and K � ker(ϵ) is module with fpr and pd R (K) ≤ n − 1.
Proof.Since M has a fpr, we can consider an exact sequence where all P i is fnitely generated projective modules and N is a module with fpr.According to ([4], Teorem 2.20), N is G-projective.Using Lemma 11, we obtain an exact sequence: where all Q i is fnitely generated projective and G is a Gprojective module with fpr, and such that the functor Hom(− , Q) maintains the exactness of this sequence when Q is projective.Tis enables the construction of homomorphisms Q i ⟶ P n− 1− i for i � 0, . . ., n − 1 and G ⟶ M such that the following diagram is commutative.
resulting in the exactness of the sequence It is worth noting that P 0 ⊕ G has a fpr.Consequently, the kernel K of ϵ: P 0 ⊕ G ⟶ M satisfes pd R (K) ≤ n − 1 and has a fpr (by ([8], Teorem 2.1.2)).

Corollary 7 .( 1 )Remark 8 .
If R is a domain, then the following are equivalent: R is a Dedekind domain.(2)R is Noetherian, fPD(R) ≤ 1,and R has a fnite global dimension.(3) R is Noetherian regular DW domain.Recall the classical defnition of regularity for Noetherian rings: A local Noetherian ring R is regular if it has a fnite global dimension.
□Example 2. Let k be a feld.Consider the additive group