w-FP-projective Modules and Dimensions

Abstract

Let R be  a ring. An R-module M is said to be an absolutely w-pure module if and only if $\text {Ext}^1_R(F,M)$ is a GV-torsion module for any finitely presented module F. In this paper, we introduce and study the concept of w-FP-projective module which is in some way a generalization of the notion of $\textrm{FP}$-projective module. An R-module M is said to be w-$\textrm{FP}$-projective if $\text {Ext}^1_R(M,N)=0$ for any absolutely w-pure module N. This new class of modules will be used to characterize (Noetherian) DW rings. Hence, we introduce the w-$\textrm{FP}$-projective dimensions of modules and rings. The relations between the introduced dimensions and other (classical) homological dimensions are discussed. Illustrative examples are given.

1 Introduction

Throughout, all rings considered are commutative with unity and all modules are unital. Let R be a ring and M be an R-module. As usual, we use $\textrm{pd}_R(M)$, $\textrm{id}_R(M)$, and $\textrm{fd}_R(M)$ to denote, respectively, the classical projective dimension, injective dimension, and flat dimension of M, and $\textrm{wdim}(R)$ and $\textrm{gldim}(R)$ to denote, respectively, the weak and global homological dimensions of R.

Now, we review some definitions and notation. Let J be an ideal of R. Following [9], J is called a Glaz-Vasconcelos ideal (a GV-ideal for short) if J is finitely generated and the natural homomorphism $\varphi : R \rightarrow J^{*} = \textrm{Hom}_R(J,R)$ is an isomorphism. Let M be an R-module and define

$$\textrm{tor}_{GV}(M) = \{x\in M \mid Jx = 0\text { for some } J\in GV(R)\},$$

where GV(R) is the set of GV-ideals of R. It is clear that $\textrm{tor}_{GV}(M)$ is a submodule of M. Now M is said to be GV-torsion (resp., GV-torsion-free) if $\textrm{tor}_{GV}(M) =M$ (resp., $\textrm{tor}_{GV}(M) =0$). A GV-torsion-free module M is called a w-module if $\textrm{Ext}^1_R(R/J, M) =0$ for any $J\in GV(R)$. Projective modules and reflexive modules are w-modules. In the recent paper [17], it was shown that flat modules are w-modules. The notion of w-modules was introduced firstly over a domain [16] in the study of Strong Mori domains and was extended to commutative rings with zero divisors in [9]. Let w-$\textrm{Max}(R)$ denote the set of maximal w-ideals of R, i.e., w-ideals of R maximal among proper integral w-ideals of R. Following [9, Proposition 3.8], every maximal w-ideal is prime. For any GV-torsion-free module M,

$$M_{w}:=\{x\in E(M)\mid Jx\subseteq M \text { for some } J\in GV(R)\}$$

is a w-submodule of E(M) containing M and is called the w-envelope of M, where E(M) denotes the injective hull of M. It is clear that a GV-torsion-free module M is a w-module if and only if $M_{w}=M$. Let M and N be R-modules and let $f : M \rightarrow N$ be a homomorphism. Following [18], f is called a w-monomorphism (resp., w-epimorphism, w-isomorphism) if $f_{\mathfrak {p}} : M_{\mathfrak {}}\rightarrow N_{\mathfrak {p}}$ is a monomorphism (resp., an epimorphism, an isomorphism) for all $\mathfrak {p}\in w\text {-}\mathrm{{Max}}(R)$. A sequence $0\rightarrow A \rightarrow B \rightarrow C\rightarrow 0$ of R-modules is said to be w-exact if Recall from [12] that an R-module A is called absolutely pure if A is a pure submodule in every R-module which contains A as a submodule. C. Megibben showed in [20], that an R-module A is absolutely pure if and only if ${\text {Ext}}_R^1(N,A)=0$ for every finitely presented R-module N. Hence, an absolutely pure module is precisely an FP-injective module in [21]. For more details about absolutely pure (or FP-injective) modules, see [3, 12, 19,20,21]. In a very recent paper[4], the authors introduced the notion of absolutely w-pure modules as generalization of absolutely pure (FP-injective) modules in the sense of the w-operation theory. As in [5], a w-exact sequence of R-modules $0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$ is said to be w-pure exact if, for any R-module M, the induced sequence $0\rightarrow A\otimes M\rightarrow B\otimes M \rightarrow C\otimes M \rightarrow 0$ is w-exact. In particular, if A is a submodule of B and $0\rightarrow A\rightarrow B\rightarrow B/A\rightarrow 0$ is a w-pure exact sequence of R-modules, then A is said to be a w-pure submodule of B. If A is a w-pure submodule in every R-module which contains A as a submodule, then A is said to be an absolutely w-pure module. Following [4, Theorem 2.6], an R-module A is absolutely w-pure if and only if ${\text {Ext}}_R^1(N,A)$ is a GV-torsion R-module for every finitely presented R-module N. In [1], Ding and Mao introduced and studied the notion of $\textrm{FP}$-projective dimension of modules and rings; the $\textrm{FP}$-projective dimension of an R-module M, denoted by $\textrm{fpd}_{R}(M)$, is the smallest positive integer n for which ${\text {Ext}}_R^{n+1}(M,A)=0$ for all absolutely pure ($\textrm{FP}$-injective) R-modules A, and $\textrm{FP}$-projective dimension of R, denoted by $\textrm{fpD}(R)$, is defined as the supremum of the $\textrm{FP}$-projective dimensions of finitely generated R-modules. These dimensions measure how far away a finitely generated module is from being finitely presented, and how far away a ring is from being Noetherian.

In Sect. 2, we introduce the concept of w-FP-projective modules. Hence, we prove that a ring R is DW ([14]) if and only if every FP-projective R-module is w-FP-projective if and only if every finitely presented R-module is w-FP-projective, and R is a coherent DW-ring if and only if every finitely generated ideal is w-FP-projective.

Section 3 deals with the w-FP projective dimension of modules and rings. After a routine study of these dimensions, we prove that R is a Noetherian DW-ring if and only if every R-module is w-FP-projective and R is FP-hereditary DW-ring if and only if every submodule of projective R-module is w-FP-projective.

2 W-FP-projective Modules

We start with the following definition.

Definition 1

An R-module M is said to be w-$\textrm{FP}$-projective if ${\text {Ext}}_R^1(M,A)=0$ for any absolutely w-pure R-module A.

Since every absolutely pure module is absolutely w-pure ([4, Corollary 2.7]), we have the following inclusions:

$$\{\text {Projective modules}\}\subseteq \{w\text {-FP-projective modules}\}\subseteq \{\text {FP-projective modules}\}$$

Recall that a ring R is called a DW-ring if every ideal of R is a w-ideal, or equivalently every maximal ideal of R is w-ideal [14]. Examples of DW-rings are Prüfer domains, domains with Krull dimension one, and rings with Krull dimension zero. Hence, it is clear that if R is a DW-ring, then w-$\textrm{FP}$-projective R-modules are just the $\textrm{FP}$-projective R-modules. Moreover, using [4, Corollary 2.9], it is easy to see that over a von Neumann regular ring, the three classes of modules above coincide.

Remark 1

It is proved in [15] that a finitely generated R-module M is finitely presented if and only if ${\text {Ext}}_{R}^1(M,A)=0$ for any absolutely pure ($\textrm{FP}$-injective) R-module A. Thus, every finitely generated $w\text {-}\mathrm{{FP}}$-projective R-module is finitely presented.

We need the following lemma.

Lemma 1

Every GV-torsion R-module is absolutely w-pure.

Proof

Let A be an arbitrary R-module and N be a finitely presented R-module. For any maximal w-ideal $\mathfrak {p}$ of R, the naturel homomorphism

$$ \theta :\text {Hom}_{R}(N,A)_{\mathfrak {p}}\rightarrow \text {Hom}_{R_{\mathfrak {p}}}(N_{\mathfrak {p}},A_{\mathfrak {p}})$$

induces a homomorphism

$$ \theta _{1}: {\text {Ext}}_{R}^1(N,A)_{\mathfrak {p}}\rightarrow {\text {Ext}}_{R_{\mathfrak {p}}}^1(N_{\mathfrak {p}},A_{\mathfrak {p}})$$

Following [7, Proposition 1.10], $\theta _{1}$ is a monomorphism. Suppose that A is a GV-torsion R-module. Then, we get $({\text {Ext}}_{R}^1(N,A))_{\mathfrak {p}} = 0$ since $A_{\mathfrak {p}}=0$ (by [7, Lemma 0.1]). Hence, ${\text {Ext}}_{R}^1(N,A)$ is GV-torsion (by [7, Lemma 0.1]). Consequently, A is an absolutely w-pure R-module (by [4, Theorem 2.6]).   $\square$

The first main result of this paper characterizes DW-rings in terms of w-$\textrm{FP}$-projective R-modules.

Proposition 1

Let R be a ring. Then the following conditions are equivalent:

• (1) Every finitely presented R-module is w-$\textrm{FP}$-projective.

• (2) Every $\textrm{FP}$-projective R-module is w-$\textrm{FP}$-projective.

• (3) R is a DW-ring.

Proof

$(3)\Rightarrow (2)$ is obvious and $(2)\Rightarrow (1)$ follows from the fact that finitely presented R-modules are always FP-projective.

$(1)\Rightarrow (3)$ Suppose that R is not a DW-ring. Then, by [8, Theorem 6.3.12], there exist maximal ideal $\mathfrak {m}$ of R which is not w-ideal, and so by [8, Theorem 6.2.9], $\mathfrak {m}_{w}=R$. Hence, by [8, Proposition 6.2.5], $R/\mathfrak {m}$ is a GV-torsion R-module (sine $\mathfrak {m}$ is a GV-torsion-free R-module), and so $R/\mathfrak {m}$ is an absolutely w-pure R-module (by Lemma 1). Hence, by hypothesis, for any I finitely generated ideal I of R, we get ${\text {Ext}}_{R}^1(R/I,R/\mathfrak {m})=0$ since R/I is a finitely presented R-module. Using [10, Lemma 3.1], we obtain that ${\text {Tor}}_{R}^1(R/I,R/\mathfrak {m})=0$, which means that $R/\mathfrak {m}$ is flat. Accordingly, $\mathfrak {m}$ is a w-ideal, and then $\mathfrak {m}_{w}=\mathfrak {m}$, a contradiction with $\mathfrak {m}_{w} = R$. Consequently, R is a DW-ring.   $\square$

Next, we will give an example of $\textrm{FP}$-projective module, which is not w-$\textrm{FP}$-projective.

Example 1

Let $(R, \mathfrak {m})$ be a regular local ring with $\textrm{gldim}(R) =n$ ($n\ge 2$). By [2, Example 2.6], R is not DW ring. Hence, there exists an $\textrm{FP}$-projective R-module M which is not w-$\textrm{FP}$-projective.

Next, we give some characterizations of w-$\textrm{FP}$-projective modules.

Proposition 2

Let M be an R-module. Then the following are equivalent:

• 1. M is w-$\textrm{FP}$-projective.

• 2. M is projective with respect to every exact sequence $0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$, where A is absolutely w-pure.

• 3. $P\otimes M$ is w-$\textrm{FP}$-projective for any projective R-module P.

• 4. $\text {Hom}(P,M)$ is w-$\textrm{FP}$-projective for any finitely generated projective R-module P.

Proof

$(1)\Leftrightarrow (2)$ is straightforward.

$(1)\Rightarrow (3)$ Let A be any absolutely w-pure R-module and P be a projective R-module. Following [8, Theorem 3.3.10], we have the isomorphism:

$${\text {Ext}}_{R}^1(P\otimes M,A)\cong \text {Hom}(P,{\text {Ext}}_{R}^1(M,A)).$$

Since M is w-$\textrm{FP}$-projective, we have ${\text {Ext}}_{R}^1(M,A)=0$. Thus, ${\text {Ext}}_{R}^1(P\otimes M,A)=0$, and so $P\otimes M$ is w-$\textrm{FP}$-projective.

$(1)\Rightarrow (4)$ Let A be any absolutely w-pure R-module and P be a finitely generated projective R-module. Using [8, Theorem 3.3.12], we have the isomorphism:

$${\text {Ext}}_{R}^1(\text {Hom}(P,M),A)\cong P\otimes {\text {Ext}}_{R}^1(M,A)=0.$$

Hence, $\text {Hom}(P,M)$ is a w-$\textrm{FP}$-projective R-module.

$(3)\Rightarrow (1)$ and $(4)\Rightarrow (1)$ Follow by letting $P=R$.   $\square$

Recall that a fractional ideal I of a domain R is said to be w-invertible if $(II^{-1})_{w}=R$. A domain R is said to be a Prüfer v-multiplication domain (PvMD) when any nonzero finitely generated ideal of R is w-invertible. Equivalently, R is a PvMD if and only if $R_{\mathfrak {p}}$ is a valuation domain for any maximal w-ideal $\mathfrak {p}$ of R ([23, Theorem 2.1]). The class of PvMDs strictly contains the classes of Prüfer domains, Krull domains, and integrally closed coherent domains.

Proposition 3

Let R be a PvMD. Then $\textrm{pd}_{R}(M)\le 1$ for any w-$\textrm{FP}$-projective R-module M.

Proof

Let M be a w-$\textrm{FP}$-projective R-module. Following [4, Theorem 2.10], every h-divisible R-module is absolutely w-pure. Hence, ${\text {Ext}}_{R}^1(M,D)=0$ for any h-divisible R-module D. Hence, by [22, vii, Proposition 2.5], $\textrm{pd}_{R}(M)\le 1$, as desired.   $\square$

Proposition 4

If M is a w-$\textrm{FP}$-projective R-module and ${\text {Ext}}_{R}^1(M,G)=0$ for any GV-torsion-free R-module G, then M is projective.

Proof

Let A be an arbitrary R-module. The exact sequence $0\rightarrow \textrm{tor}_{GV}(A)\rightarrow A\rightarrow A/\textrm{tor}_{GV}(A)\rightarrow 0$ gives rise to the exact sequence $0{=}{\text {Ext}}_{R}^1(M,\textrm{tor}_{GV}(A))\rightarrow {\text {Ext}}_{R}^1(M,A)\rightarrow {\text {Ext}}_{R}^1(M, A/\textrm{tor}_{GV}(A)){=}0$ Thus ${\text {Ext}}_{R}^1(M,A)=0$, and so M is projective.   $\square$

Proposition 5

Let $(R,\mathfrak {m})$ be a local ring which is not DW-ring (for example, regular local rings R with $\textrm{gldim}(R) =n$ ($n\ge 2$)). Then every finitely generated w-$\textrm{FP}$-projective R-module M is free.

Proof

Let M be a finitely generated w-$\textrm{FP}$-projective R-module. As in the proof of Proposition 1, there exist a maximal ideal $\mathfrak {m}$ of R which is not w-ideal, and so $R/\mathfrak {m}$ is an absolutely w-pure R-module. we obtain that ${\text {Tor}}_{R}^1(M,R/\mathfrak {m})=0$. But M is finitely generated, and so finitely presented (by Remark 1). Hence, by [11, Lemma 2.5.8], M is projective. Consequently, M is free since R is local.   $\square$

Proposition 6

The class of all $w\text {-}\mathrm{{FP}}$-projective modules is closed under arbitrary direct sums and under direct summands.

Proof

Follows from [8, Theorem 3.3.9(2)].   $\square$

Recall that a ring R is called coherent if every finitely generated ideal of R is finitely presented.

Lemma 2

Let R be a coherent ring and A be an R-module. Then A is absolutely w-pure if and only if ${\text {Ext}}_{R}^{n+1}(N,A)$ is a GV-torsion R-module for any finitely presented module N and any integer $n\ge 0$.

Proof

$(\Rightarrow )$ suppose that A is absolutely w-pure R-module and let N be a finitely presented R-module. The case $n=0$ is obvious. Hence, assume that $n> 0$. Consider an exact sequence

$$0\rightarrow N' \rightarrow F_{n-1}\rightarrow \cdots \rightarrow F_{0}\rightarrow N\rightarrow 0$$

where $F_{0}$,...,$F_{n-1}$ are finitely generated free R-modules and $N'$ is finitely presented. Such sequence exists since R is coherent. Thus, $({\text {Ext}}_{R}^{n+1}(N,A))_{\mathfrak {p}}\cong ({\text {Ext}}_{R}^{1}(N',A))_{\mathfrak {p}}=0$ for any w-maximal ideal $\mathfrak {p}$ of R. So, ${\text {Ext}}_{R}^{n+1}(N,A)$ is a GV-torsion R-module.

$(\Leftarrow )$ Clear.   $\square$

Lemma 3

Let R be a coherent ring and $0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$ be an exact sequence of R-modules, where A is absolutely w-pure. Then, B is absolutely w-pure if and only if C is absolutely w-pure.

Proof

Let N be a finitely presented R-module. We have${\text {Ext}}_{R}^{1}(N,A)\rightarrow {\text {Ext}}_{R}^{1}(N,B)\rightarrow {\text {Ext}}_{R}^{1}(N,C)\rightarrow {\text {Ext}}_{R}^{2}(N,A)$ By Lemma 2, for any maximal w-ideal $\mathfrak {p}$, we get $0={\text {Ext}}_{R}^{1}(N,A)_{\mathfrak {p}}\rightarrow {\text {Ext}}_{R}^{1}(N,B)_{\mathfrak {p}}\rightarrow {\text {Ext}}_{R}^{1}(N,C)_{\mathfrak {p}}\rightarrow {\text {Ext}}_{R}^{2}(N,A)_{\mathfrak {p}}=0.$ Thus, ${\text {Ext}}_{R}^{1}(N,B)_{\mathfrak {p}}\cong {\text {Ext}}_{R}^{1}(N,C)_{\mathfrak {p}}$. So, ${\text {Ext}}_{R}^{1}(N,B)$ is a GV-torsion R-module if and only if ${\text {Ext}}_{R}^{1}(N,C)$ is a GV-torsion R-module. Thus, B is absolutely w-pure if and only if C is absolutely w-pure.   $\square$

Proposition 7

Let R be a coherent ring and M be an R-module. Then the following are equivalent:

• 1. M is w-$\textrm{FP}$-projective.

• 2. ${\text {Ext}}_{R}^{n+1}(M,A)=0$ for any absolutely w-pure module A and any integer $n\ge 0$.

Proof

$(1)\Rightarrow (2)$ Let A be an absolutely w-pure R-module. The case $n=0$ is obvious. So, we may assume $n>0$. Consider an exact sequence

$$0\rightarrow A \rightarrow E^{0}\rightarrow \cdots \rightarrow E^{n-1}\rightarrow A^{'}\rightarrow 0$$

where $E^{0}$,...,$E^{n-1}$ are injective R-modules. By Lemma 3, $A'$ is absolutely w-pure. Hence, ${\text {Ext}}_{R}^{n+1}(M,A)\cong {\text {Ext}}_{R}^{1}(M,A')=0$.

$(2)\Rightarrow (1)$ Obvious.   $\square$

Proposition 8

Let R be a coherent ring and $0\rightarrow M''\rightarrow M'\rightarrow M\rightarrow 0$ be an exact sequence of R-modules, where M is w-$\textrm{FP}$-projective. Then, $M'$ is w-$\textrm{FP}$-projective if and only if $M''$ is w-$\textrm{FP}$-projective.

Proof

Follows from Proposition 7.   $\square$

We end this section with the following characterizations of a coherent DW-rings.

Proposition 9

Let R be a ring. Then the following are equivalent:

• 1. R is a coherent DW-ring.

• 2. Every finitely generated submodule of a projective R-module is w-$\textrm{FP}$-projective.

• 3. Every finitely generated ideal of R is w-$\textrm{FP}$-projective.

Proof

$(1)\Rightarrow (2)$ Follows immediately from [13, Theorem 3.7] since, over a DW-ring, the classes of w-$\textrm{FP}$-projective modules and $\textrm{FP}$-projective modules coincide.

$(2)\Rightarrow (3)$ Obvious.

$(3)\Rightarrow (1)$ R is coherent by Remark 1. Assume that R is not a DW-ring. As in the proof of Proposition 1, there exist a maximal ideal $\mathfrak {m}$ of R such that $R/\mathfrak {m}$ is absolutely w-pure and $\mathfrak {m}_{w}=R$. So, for any finitely generated ideal I of R, we have

$$0={\text {Ext}}_{R}^1(I,R/\mathfrak {m})\rightarrow {\text {Ext}}_{R}^2(R/I,R/\mathfrak {m}) \rightarrow {\text {Ext}}_{R}^2(R,R/\mathfrak {m})=0,$$

and then ${\text {Ext}}_{R}^2(R/I,R/\mathfrak {m})=0$. By [10, Lemma 3.1], ${\text {Tor}}_{R}^2(R/I,R/\mathfrak {m})=0$, which means that $\textrm{fd}_{R}(R/\mathfrak {m})\le 1$. Then $\mathfrak {m}$ is flat, and so a w-ideal, a contradiction.   $\square$

Corollary 1

Let R be a domain. Then R is a coherent DW-domain if and only if every finitely generated torsion-free R-module is w-$\textrm{FP}$-projective.

Proof

Following [8, Theorem 1.6.15], every finitely generated torsion-free R-module can be embedded in a finitely generated free module (since R is a domain). Hence, $(\Rightarrow )$ follows immediately from Proposition 9. For $(\Leftarrow )$, it suffices to see that since R is a domain, every ideal is torsion-free, and then use Proposition 9.   $\square$

3 The W-FP-projective Dimension of Modules and Rings

In this section, we introduce and investigate the w-FP-projective dimension for modules and rings.

Definition 2

Let R be a ring. For any R-module M, the w-$\textrm{FP}$-projective dimension of M, denoted by w-$\textrm{fpd}_R(M)$, is the smallest integer $n\ge 0$ such that ${\text {Ext}}_R^{n+1}(M,A)=0$ for any absolutely w-pure R-module A. If no such integer exists, set w-$\textrm{fpd}_R(M)=\infty$.

The w-FP-projective dimension of R is defined by

$$w\text {-}\mathrm{{fpD}}(R)=\sup \{w\text {-}\mathrm{{fpD}}_R(M) :\text { M is finitely generated } R\text {-module}\}$$

Clearly, an R-module M is w-$\textrm{FP}$-projective if and only if $w\text {-fpd}_R(M)=0$, and $\textrm{fpd}_R(M)\le w\text {-fpd}_R(M)$, with equality when R is a DW-ring. However, this inequality may be strict (Remark 1). Also, $\text {fpD}(R)\le w\text {-}\mathrm{{fpD}}(R)$ with equality when R is a DW-ring, and this inequality may be strict. To see that, consider a regular local ring $(R, \mathfrak {m})$ with $\textrm{gldim}(R) =n$ ($n\ge 2$). Since R is Noetherian, we get $\textrm{fpD}(R)=0$ (by [1, Proposition 2.6]). Moreover, by Remark 1, there exists an (FP-projective) R-module M which is not w-FP-projective. Thus, $w\text {-fpD}(R)>0$.

First, we give a description of the w-FP-Projective dimension of modules over coherent ring.

Proposition 10

Let R be a coherent ring. The following statements are equivalent for an R-module M.

• 1. $w\text {-}\mathrm{{fpd}}(M)\leqslant n$.

• 2. ${\text {Ext}}_R^{n+1}(M,A)=0$ for any absolutely w-pure R-module A.

• 3. ${\text {Ext}}_R^{n+j}(M,A)=0$ for any absolutely w-pure R-module A and any $j\ge 1$.

• 4. If the sequence $0\rightarrow P_n\rightarrow P_{n-1}\rightarrow \cdots \rightarrow P_0\rightarrow M\rightarrow 0$ is exact with $P_0,\ldots , P_{n-1}$ are $w\text {-}\mathrm{{FP}}$-projective R-modules, then $P_n$ is $w\text {-FP}$-projective.

• 5. If the sequence $0\rightarrow P_n\rightarrow P_{n-1}\rightarrow \cdots \rightarrow P_0\rightarrow M\rightarrow 0$ is exact with $P_0,\ldots , P_{n-1}$ are projective R-modules, then $P_n$ is $w\text {-}\mathrm{{FP}}$-projective.

• 6. There exists an exact sequence $0\rightarrow P_n\rightarrow P_{n-1}\rightarrow \cdots \rightarrow P_0\rightarrow M\rightarrow 0$ where each $P_{i}$ is $w\text {-FP}$-projective.

Proof

$(3)\Rightarrow (2)\Rightarrow (1)$ and $(4)\Rightarrow (5)\Rightarrow (6)$ are trivial.

$(1)\Rightarrow (4)$ Let $0\rightarrow P_n\rightarrow P_{n-1}\rightarrow \cdots \rightarrow P_0\rightarrow M\rightarrow 0$ be an exact sequence of R-modules with $P_{0}, \ldots , P_{n-1}$ are $w\text {-FP}$-projective, and set $K_{0}=\text {Ker}(P_{0}\rightarrow M)$ and $K_{i}=\text {Ker}(P_{i}\rightarrow P_{i-1})$, where $i=1,\ldots ,n-1$. Using Proposition 7, we get

$$0={\text {Ext}}_R^{n+1}(M,A)\cong {\text {Ext}}_R^{n}(K_0,A) \cong \cdots \cong {\text {Ext}}_R^{1}(P_{n},A)$$

for all absolutely w-pure R-module A. Thus, $P_{n}$ is $w\text {-FP}$-projective.

$(6)\Rightarrow (3)$ We procced by induction on $n\ge 0$. For the $n=0$, M is $w\text {-FP}$-projective module and so (3) holds by proposition 7. If $n\ge 1$, then there is an exact sequence $0\rightarrow P_n\rightarrow P_{n-1}\rightarrow \cdots \rightarrow P_0\rightarrow M\rightarrow 0$ where each $P_{i}$ is $w\text {-FP}$-projective. Set $K_{0}=\text {Ker}(P_{0}\rightarrow M)$. Then, we have the following exact sequences

$$0\rightarrow P_n\rightarrow P_{n-1}\rightarrow \cdots \rightarrow P_{1}\rightarrow K_{0}\rightarrow 0$$

and

$$0\rightarrow K_{0}\rightarrow P_{0}\rightarrow M\rightarrow 0$$

Hence, by induction ${\text {Ext}}_R^{n-1+j}(K_{0},A)=0$ for all absolutely w-pure R-module A and all $j\ge 1$. Thus, ${\text {Ext}}_R^{n+j}(M,A)=0$, and so we have the desired result.   $\square$

The proof of the next proposition is standard homological algebra. Thus we omit its proof.

Proposition 11

Let R be a coherent ring and $0\rightarrow M''\rightarrow M'\rightarrow M\rightarrow 0$ be an exact sequence of R-modules. If two of $w\text {-fpd}_R(M'')$, $w\text {-fpd}_R(M')$ and $w\text {-fpd}_R(M)$ are finite, so is the third. Moreover

• 1. $w\text {-fpd}_R(M'')\le \sup \left\{ w\text {-fpd}_R(M'),\;w\text {-fpd}_R(M)-1\right\}$.

• 2. $w\text {-fpd}_R(M')\le \sup \{w\text {-fpd}_R(M''),\;w\text {-fpd}_R(M)\}$.

• 3. $w\text {-fpd}_R(M)\le \sup \{w\text {-fpd}_R(M'),\;w\text {-fpd}_R(M'')+1\}$.

Corollary 2

Let R be a coherent ring and $0\rightarrow M''\rightarrow M'\rightarrow M\rightarrow 0$ be an exact sequence of R-modules. If $M'$ is w-$\textrm{FP}$-projective and w-$\textrm{fpd}_R(M)>0$, then $w\text {-fpd}_R(M)=w\text {-fpd}_R(M'')+1$.

Proposition 12

Let R be a coherent ring and $\{M_{i}\}$ be a family of R-modules. Then $w\text {-fpd}_R(\oplus _{i}M_{i})=\sup _{i}\{w\text {-fpd}_R(M_{i})\}$.

Proof

The proof is straightforward.   $\square$

Proposition 13

Let R be a ring and $n\ge 0$ be an integer. Then the following statements are equivalent:

• 1. $w\text {-fpD}(R)\le n$.

• 2. $w\text {-fpd}(M)\leqslant n$ for all R-modules M.

• 3. $w\text {-fpd}(R/I)\leqslant n$ for all ideals I of R.

• 4. $id_{R}(A)\leqslant n$ for all absolutely w-pure R-modules A.

Consequently, we have

$$\begin{aligned} w\text {-fpD}(R)&=\sup \{w\text {-fpd}_R(M) \mid M~\text {is an}~R\text {-module}\}\\&=\sup \{w\text {-fpd}_R(R/I) \mid I~\text {is an ideal of}~R \}\\&=\sup \{id_R(A) \mid A~\text {is an abosolutely}~w\text {-pure}~R\text {-module}\} \end{aligned}$$

Proof

$(2)\Rightarrow (1)\Rightarrow (3)$ are trivial.

$(3)\Rightarrow (4)$ Let A be an absolutely w-pure R-module. For any ideal I of R, we have ${\text {Ext}}_R^{n+1}(R/I,A)=0$. Thus, $id_{R}(A)\leqslant n$.

$(4)\Rightarrow (2)$ Let M be an R-module. For any absolutely w-pure R-module A, we have ${\text {Ext}}_R^{n+1}(M,A)=0$. Hence, $w\text {-fpd}(M)\leqslant n$.   $\square$

Note that Noetherian rings need not to be DW (for example, a regular ring with global dimension 2), and DW-rings need not to be Noetherian (for example, a non-Noetherian von Neumann regular ring). Next, we show that rings R with $w\text {-fpD}(R)=0$ are exactly Noetherian DW-rings.

Proposition 14

Let R be a ring. Then the following are equivalent:

• 1. $w\text {-fpD}(R)=0$.

• 2. Every R-module is w-$\textrm{FP}$-projective.

• 3. R/I is w-$\textrm{FP}$-projective for every ideal I of R.

• 4. Every absolutely w-pure R-module is injective.

• 5. R is Noetherian DW-ring.

Proof

The equivalence of (1), (2), (3), and (4) follows from Proposition 13.

$(2)\Leftrightarrow (5)$ Follows from Proposition 1 and [1, Proposition 2.6].   $\square$

Recall from[13], that a ring R is said FP-hereditary if every ideal of R is FP-projective. Note that FP-hereditary rings need not to be DW (for example, a non DW Noetherian ring), and DW-rings need not to be FP-hereditary (for example, a non-Noetherian von Neumann regular ring). Next, we show that rings R with $w\text {-fpD}(R)\le 1$ are exactly FP-hereditary DW-rings.

Proposition 15

Let R be a ring. Then the following are equivalent:

• 1. $w\text {-fpD}(R)\le 1$.

• 2. Every submodule of w-$\textrm{FP}$-projective R-module is w-$\textrm{FP}$-projective.

• 3. Every submodule of projective R-module is w-$\textrm{FP}$-projective.

• 4. I is w-$\textrm{FP}$-projective for every ideal I of R.

• 5. $id_{R}(A)\le 1$ for all absolutely w-pure R-module A.

• 6. R is a (coherent) FP-hereditary DW-ring.

Proof

The implications $(2)\Rightarrow (3)\Rightarrow (4)$ are obvious.

$(4)\Rightarrow (5)$ Let A be an absolutely w-pure R-module and I be an ideal of R. The exact sequence $0\rightarrow I\rightarrow R\rightarrow R/I\rightarrow 0$ gives rise to the exact sequence

$$0={\text {Ext}}_R^{1}(I,A)\rightarrow {\text {Ext}}_R^{2}(R/I,A)\rightarrow {\text {Ext}}_R^{2}(R,A)=0.$$

Thus, ${\text {Ext}}_R^{2}(R/I,A)=0$, and so $id_{R}(A)\le 1$.

$(5)\Rightarrow (4)$ Let I be an ideal of R. For any absolutely w-pure R-module A, we have

$$0={\text {Ext}}_R^{2}(R/I,A)={\text {Ext}}_R^{1}(I,A).$$

Thus, I is w-$\textrm{FP}$-projective.

$(4)\Rightarrow (6)$ By hypothesis, R is FP-hereditary. Now, by Proposition 9 R is a coherent DW-ring.

$(6)\Rightarrow (2)$ By [13, Theorem 3.16], since the w-$\textrm{FP}$-projective R-modules are just the $\textrm{FP}$-projective R-modules over a DW-ring.

$(1)\Leftrightarrow (5)$ By Proposition 13.   $\square$

Remark 2

In the Example 1, the ring R is coherent but not DW. Then, R contains a finitely generated ideal which is not w-$\textrm{FP}$-projective by Proposition 5.