1 Introduction

Throughout this paper, R is always a commutative ring with \(0\ne 1\). For an R-module M, \(\text{ pd }_{R}M\) denotes, as usual, the projective dimension of M. The dual module \(\mathrm{Hom}_{R}(M,R)\) of M is always denoted by \(M^{*}\). It is well known that an R-module M is called FP-injective in [28] if \(\mathrm{Ext}_{R}^{1}(F,M)=0\) for every finitely presented R-module F. Recall that an R-module M is called super finitely presented in [16] if there exists an exact sequence of R-modules

$$\begin{aligned} \cdots \rightarrow P_{n}\rightarrow \cdots \rightarrow P_{1}\rightarrow P_{0}\rightarrow M\rightarrow 0 \end{aligned}$$

such that every \(P_{i}\) is finitely generated and projective. The super finitely presented module originated from Grothendieck’s notion of a pseudo-coherent module in [3]; in [4,5,6, 20], the authors used the term “\(\mathrm{FP}_{\infty }\)-module” in the sense of a super finitely presented module; and in [1], Bennis called it an infinitely presented module. It is obvious that every super finitely presented module is finitely presented. In the Gorenstein homological algebra, the super finitely presented module might be an important tool instead of the finitely presented module. For example, in [16], Gao and Wang make use of the super finitely presented modules to study the Mahdou–Tamekkante–Yassemi’s question ([24, p. 436]), and they prove in [16, Theorem 2.6] that a ring R is a Gorenstein semi-hereditary ring if and only if every finitely generated submodule of a projective R-module is Gorenstein projective. Bennis proves in [1, Theorem 3.3] that every super finitely presented Gorenstein flat module is Gorenstein projective. Recently, the weak injective modules were introduced in [17] as a generalization of the FP-injective modules. We recall that an R-module M is called weak injective if \(\mathrm{Ext}_{R}^{1}(N,M)=0\) for every super finitely presented R-module N. In [5], a weak injective module is also called an absolutely clean module. The term “absolutely clean module” is developed from Maddox’s notion of an absolutely pure submodule in [22], where an absolutely pure submodule is precisely an FP-injective module (See [25, Proposition 1]). Accordingly, the weak injective dimension of M, denoted by \(\text{ wid }_{R}M\), is defined to be the smallest \(n\ge 0\) such that \(\mathrm{Ext}_{R}^{n+1}(N,M)=0\) for all super finitely presented R-modules N (if no such n exists, set \(\text{ wid }_{R}M=\infty \)). Also, in [17], the authors study the super finitely presented dimension of a ring R, which is defined as

$$\begin{aligned} \text{ s.gl.dim }(R)=\sup \{\text{ pd }_{R}M\,|\, M\,\text{ is } \text{ a } \text{ super } \text{ finitely } \text{ presented }\,R\text{-module }\}. \end{aligned}$$

In this paper, we shall study the Gorenstein super finitely presented dimension of a ring and investigate two questions which are closely related to it.

Now recall that an R-module M is called Gorenstein projective in [10] if there exists an exact sequence of projective R-modules \(\cdots \rightarrow P_{1}\rightarrow P_{0}\rightarrow P^{0}\rightarrow P^{1}\rightarrow \cdots \) such that \(M\cong \mathrm{ker}(P_{0}\rightarrow P^{0})\) and the functor \(\mathrm{Hom}(-,Q)\) leaves the sequence exact whenever Q is a projective R-module. An R-module M is called Gorenstein flat in [11] if there exists an exact sequence \(\dots \rightarrow F_{1}\rightarrow F_{0}\rightarrow F^{0}\rightarrow F^{1}\rightarrow \cdots \) of flat R-modules such that \(M\cong \mathrm{ker}(F_{0}\rightarrow F^{0})\) and the functor \(E\mathop {\otimes }\limits _{R}-\) leaves the sequence exact whenever E is an injective R-module. As in [18], the Gorenstein projective and flat dimensions of R-module M are defined in terms of Gorenstein projective and flat resolutions, denoted by \(\text{ Gpd }_{R}M\) and \(\text{ Gfd }_{R}M\), respectively. In [2], the Gorenstein global dimension of a ring R is defined as

$$\begin{aligned} \text{ Ggl.dim }(R)=\sup \{\text{ Gpd }_{R}M\,|\, M\,\text{ is } \text{ an }\, R\text{-module }\}. \end{aligned}$$

Similarly, the weak Gorenstein global dimension of a ring R is defined as

$$\begin{aligned} \text{ wGgl.dim }(R)=\sup \{\text{ Gfd }_{R}M\,|\, M\,\text{ is } \text{ an }\, R\text{-module }\}. \end{aligned}$$

Now, the Gorenstein super finitely presented dimension of a ring R can be defined as follows:

$$\begin{aligned} G\text{- }\text{ s.gl. }\dim (R)=\sup \{\text{ Gpd }_{R}{M}\,|\,M\,\text{ is } \text{ a } \text{ super } \text{ finitely } \text{ presented }\,R\text{-module }\}. \end{aligned}$$

It is well known that a ring R is called a quasi-Frobenius (QF-ring, for short) if R is a Noetherian ring and R is self-injective. It is shown in [2, Proposition 2.6] that a ring R is a QF-ring if and only if the Gorenstein global dimension of R is zero. Also, a ring R is called a n-FC-ring in [9] if R is a coherent ring and FP-\(\text{ id }_{R}R\le n\), and the 0-FC-ring is the so-called FC-ring. It follows from [24, Proposition 2.3] that a ring R is an FC-ring if and only if the weak Gorenstein global dimension of R is zero. These kinds of “0”-dimension of rings play an important role in Gorenstein multiplicative ideal of theory now. For example, it follows from [19, Theorem 2.11] and Lemma 2.16 that a domain R is a Gorenstein Dedekind domain if and only if R / (u) is a QF-ring for any nonzero nonunit \(u\in R\), where a domain R is a Gorenstein Dedekind domain (G-Dedekind domain, for short) if every ideal of R is Gorenstein projective. It is shown in [26, Theorem 4.2] that a domain R is a Gorenstein Prüfer domain if and only if R / (u) is an FC-ring for any nonzero nonunit \(u\in R\), where a domain R is called a Gorenstein Dedekind domain (G-Prüfer domain, for short) if every finitely generated ideal of R is Gorenstein projective. So, it is natural to ask the following question,

Question 1 Whether there is a reasonable concept of “generalized FC-rings” such that it corresponds to the ring with \(G\text{- }\text{ s.gl. }\dim (R)=0\)?

We check in Proposition 2.2 that R is a coherent ring if and only if every finitely presented R-module is super finitely presented. And hence a weak injective module over a coherent ring is always FP-injective. So, we need to find a “generalized coherent ring” to study this question. We note Colby’s result in [7, Proposition 1], and it is stated that a ring R is a coherent ring if and only if \(M^{*}\) is finitely presented for any finitely presented R-module M. Naturally, a ring R can be called a generalized coherent ring (GC-ring, for short) if \(M^{*}\) is super finitely presented for any super finitely presented R-module M. And a ring R is called a generalized FC-ring (GFC-ring, for short) if R is a GC-ring and R is self-weak injective. In this paper, we shall give a series of characterizations of GFC-rings in Theorem 2.12. For Question 1, we prove in Theorem 2.12 that for a ring R, \(\text{ G.s.gl. }\dim (R)=0\) if and only if R is a GC-ring and R is self-weak injective (i.e., R is a GFC-ring). Further, in Example 2.14, we give an example to show that a GFC-ring is not necessarily an FC-ring.

Our another motivation is to study a question which is analogous to the FGF-question. Recall that a ring R is called an FGF-ring if every finitely generated R-module can be embedded in a free R-module. Faith conjecture in [12] that an FGF-ring is a QF-ring. For the research of the FGF-question, the reader can refer to [12, 14, 15, 27]. Comparing with the FGF-question, we try to consider a weaker case as follows.

Question 2 Is R a QF-ring if every (super) finitely presented R-module can be embedded in a free R-module?

For Question 2, we proved in Theorem 2.12 and Corollary 2.15 that a ring R is a GFC-ring if and only if R is a GC-ring and every super finitely presented R-module can be embedded in a free R-module; a ring R is an FC-ring if and only if R is a coherent ring and every finitely presented R-module can be embedded in a free R-module. By using this characterizations, we give an example to show that if every finitely presented R-module can be embedded in a free R-module, then R is not necessarily a QF-ring. We now proceed to state and prove our main results.

2 The Main Results

We start by the following definitions.

Definition 2.1

Let R be a ring.

  1. (1)

    R is called a generalized coherent ring (GC-ring, for short) if \(M^{*}\) is super finitely presented for any super finitely presented R-module M.

  2. (2)

    R is called an n-GFC-ring if R is a GC-ring and \(\text{ wid }_{R}(R)\le n\). In particular, if R is a 0-GFC-ring, it is said to be a GFC-ring.

Firstly, we have an observation for super finitely presented modules.

Proposition 2.2

A ring R is a coherent ring if and only if every finitely presented R-module is super finitely presented.

Proof

(\(\Rightarrow \)) Let M be a finitely presented R-module. Then, there exists a short exact sequence \(0\rightarrow K_{0}\rightarrow P_{0}\rightarrow M\rightarrow 0\), where \(K_{0}\) is finitely generated and \(P_{0}\) is finitely generated projective. Since R is a coherent ring, \(K_{0}\) is finitely presented. Hence, there exists a short exact sequence \(0\rightarrow K_{1}\rightarrow P_{1}\rightarrow K_{0}\rightarrow 0\) such that \(K_{1}\) is finitely generated and \(P_{1}\) is finitely generated projective. Continuing this process, it follows that M is super finitely presented.

(\(\Leftarrow \)) Let I be a finitely generated ideal of R. Then, \(0\rightarrow I\rightarrow R\rightarrow R/I\rightarrow 0\) is a short exact sequence. It means that R / I as an R-module is finitely presented. Hence, R / I is super finitely presented by the condition. Also since R is super finitely presented, it follows from [6, Theorem 1.8] that I is super finitely presented. So I is finitely presented. Consequently, R is a coherent ring. \(\square \)

Corollary 2.3

If R is a coherent ring, then every weak injective R-module is FP-injective.

Proof

This follows from Proposition 2.2. \(\square \)

Corollary 2.4

Every coherent ring is a GC-ring.

Proof

Let R be a coherent ring and M a super finitely presented R-module. Then, M is finitely presented. It follows from [7, Proposition 1] that \(M^{*}\) is finitely presented. Hence, \(M^{*}\) is super finitely presented by Proposition 2.2, and so R is a GC-ring. \(\square \)

Corollary 2.5

Every FC-ring is a GFC-ring.

Proof

It is clear by Corollaries 2.3 and 2.4. \(\square \)

Corollary 2.6

If R is a coherent ring, then every GFC-ring is an FC-ring.

Proof

Let R be a GFC-ring. Since R is a coherent ring, it follows from Corollary 2.3 that R is self-FP-injective. Hence, R is an FC-ring. \(\square \)

Next, we shall investigate some properties of a self-weak injective ring, and we need the following Lemma.

Lemma 2.7

Let \(F_{1}\rightarrow F_{0}\rightarrow M\rightarrow 0\) be an exact sequence of R-modules, where \(F_{0}\), \(F_{1}\) are finitely generated projective modules. Let \(N=\mathrm{cok}((F_{0})^{*}\rightarrow (F_{1})^{*})\). Then, there is an exact sequence

$$\begin{aligned} 0\rightarrow \mathrm{Ext}_{R}^{1}(N, R)\rightarrow M\rightarrow M^{**}\rightarrow \mathrm{Ext}_{R}^{2}(N, R)\rightarrow 0. \end{aligned}$$

Proof

See [21, Lemma 2.2]. \(\square \)

Theorem 2.8

The following statements hold for a ring R:

  1. (1)

    if R is self-weak injective and if an R-module M is super finitely presented such that \(M^{*}\) is also super finitely presented, then M is reflexive;

  2. (2)

    if R is a GC-ring, then R is self-weak injective if and only if every super presented R-module M is reflexive.

Proof

(1) Since M is super finitely presented, there exists a short exact sequence

$$\begin{aligned} P_{1}\rightarrow P_{0}\rightarrow M\rightarrow 0, \end{aligned}$$

where \(P_{0}\), \(P_{i}\) are finitely generated and projective. Applying the functor \(\mathrm{Hom}_{R}(-,R)\) to this sequence, we have the following exact sequence

$$\begin{aligned} 0\rightarrow M^{*}\rightarrow (P_{0})^{*}\rightarrow (P_{1})^{*}\rightarrow N\rightarrow 0 \ \ (*), \end{aligned}$$

where \(N=\mathrm{cok}((P_{0})^{*}\rightarrow (P_{1})^{*})\). By Lemma 2.7, we can get the following sequence,

$$\begin{aligned} 0\rightarrow \mathrm{Ext}_{R}^{1}(N,R)\rightarrow M \rightarrow M^{**}\rightarrow \mathrm{Ext}_{R}^{2}(N,R)\rightarrow 0. \end{aligned}$$

Since \(M^{*}\) is also super finitely presented by the condition, it is seen from the sequence \((*)\) that N is super finitely presented. Since R is self-weak injective, \(\mathrm{Ext}_{R}^{1}(N,R)=0\). On the other hand, it follows from [17, Proposition 3.1(1)] that \(\mathrm{Ext}_{R}^{2}(N,R)=0\). Consequently, \(M\cong M^{**}\) and M is reflexive.

(2) (\(\Rightarrow \)) Suppose that R is self-weak injective. Let M be a super finitely presented R-module. Then, \(M^{*}\) is super finitely presented since R is a GC-ring. By (1), M is reflexive.

(\(\Leftarrow \)) Suppose that every super finitely presented R-module is reflexive. For any super finitely presented R-module M, we need to prove \(\mathrm{Ext}_{R}^{1}(M,R)=0\). As the proof of (1), we can get the exact sequence

$$\begin{aligned} 0\rightarrow M^{*}\rightarrow (P_{0})^{*}\rightarrow (P_{1})^{*}\rightarrow N\rightarrow 0, \end{aligned}$$

where \(P_{0}\), \(P_{1}\) are finitely generated and projective. Since R is a GC-ring, \(M^{*}\) is super finitely presented. It means that N is super finitely presented. Hence, N is reflexive by the condition. From the short exact sequence

$$\begin{aligned} P_{1}\rightarrow P_{0}\rightarrow M\rightarrow 0, \end{aligned}$$

it follows that \(M\cong \mathrm{cok}((P_{1})^{**}\rightarrow (P_{0})^{**})=\mathrm{cok}(P_{1}\rightarrow P_{0})\). Hence, from the short exact sequence

$$\begin{aligned} (P_{0})^{*}\rightarrow (P_{1})^{*}\rightarrow N\rightarrow 0, \end{aligned}$$

we have the short exact sequence

$$\begin{aligned} 0\rightarrow \mathrm{Ext}_{R}^{1}(M,R)\rightarrow N\rightarrow N^{**} \end{aligned}$$

by Lemma 2.7. Since N is reflexive, it means that \(\mathrm{Ext}_{R}^{1}(M,R)=0\). So, R is self-weak injective. \(\square \)

Before we characterize the GFC-rings, we need the following results.

Lemma 2.9

Every finitely generated Gorenstein projective R-module is reflexive.

Proof

Let G be a finitely generated Gorenstein projective R-module. Then, by [16, Lemma 2.3], there exists a short exact sequence

$$\begin{aligned} 0\rightarrow G\rightarrow P\rightarrow G_{1}\rightarrow 0, \end{aligned}$$

where P is finitely generated projective, and \(G_{1}\) is finitely generated Gorenstein projective. Hence, by [18, Theorem 2.20], we have the short exact sequence \(0\rightarrow G_{1}^{*}\rightarrow P^{*}\rightarrow G^{*}\rightarrow 0\). By using the functor \(\mathrm{Hom}_{R}(-,R)\) on this short exact sequence, we get the exact sequence \(0\rightarrow G^{**}\rightarrow P^{**}\rightarrow G_{1}^{**}\rightarrow \mathrm{Ext}_{R}^{1}(G^{*},R)\rightarrow 0\). Now consider the following diagram with exact rows:

where \(\mu _{G}\) and \(\mu _{G_{1}}\) are the natural homomorphisms. Since G and \(G_{1}\) are torsionless, we have that \(\mu _{G}\) and \(\mu _{G_{1}}\) are monomorphic. Hence, \(\mu _{G}\) is isomorphic by Five Lemma. So G is reflexive. \(\square \)

Let X and F be R-modules. Consider the natural homomorphism

$$\begin{aligned} \eta _{X}:\mathrm{Hom}_{R}(X,R)\mathop {\otimes }\limits _{R}F\mapsto \mathrm{Hom}_{R}(X,F), \end{aligned}$$

where \(\eta _{X}\) is defined by \(\eta _{X}(f\otimes z)(x)=f(x)z\) for \(x\in X\), \(z\in F\) and \(f\in \mathrm{Hom}_{R}(X,R)\). It follows from [29, Theorem 2.6.14] that if X is finitely generated and projective or if X is finitely presented and F is flat, then \(\eta _{X}\) is an isomorphism. Recall that F is called weak flat in [17] if \(\mathrm{Tor}_{1}^{R}(F,N)=0\) for any super finitely module N. Now, we shall make use of the homomorphism \(\eta _{X}\) to study the weak flat modules over a GFC-ring.

Proposition 2.10

Let X be a super finitely presented R-module such that \(X^{*}\) is super finitely presented, and let F be a weak flat R-module. If \(\mathrm{Ext}_{R}^{i}(X,R)=0\) for any \(i>0\), then \(\eta _{X}\) is an isomorphism and \(\mathrm{Ext}_{R}^{i}(X,F)=0\) for all \(i>0\).

Proof

Since X is super finitely presented, we can get the following exact sequence

$$\begin{aligned} 0\rightarrow A \rightarrow P \rightarrow X\rightarrow 0, \end{aligned}$$

where P is finitely generated projective, and A is super finitely presented. Since \(\mathrm{Ext}_{R}^{1}(X,R)=0\) by the condition, we have the following exact sequence

$$\begin{aligned} 0\rightarrow X^{*} \rightarrow P^{*} \rightarrow A^{*}\rightarrow 0. \end{aligned}$$

Since \(X^{*}\) is super finitely presented by the condition, it means that \(A^{*}\) is super finitely presented. Hence, \(\mathrm{Tor}_{1}^{R}(A^{*},F)=0\) since F is weak flat. So, we have the following exact sequence

$$\begin{aligned} 0\rightarrow X^{*}\mathop {\otimes }\limits _{R}F\rightarrow P^{*}\mathop {\otimes }\limits _{R}F\rightarrow A^{*}\mathop {\otimes }\limits _{R}F\rightarrow 0. \end{aligned}$$

Now, consider the following diagram with exact rows:

It follows from the right side of above diagram that \(\eta _{X}\) is a monomorphism. Since A is also a super finitely presented R-module such that \(A^{*}\) is super finitely presented and \(\mathrm{Ext}_{R}^{1}(A,R)=0\), we can get that \(\eta _{A}\) is a monomorphism by the same argument. By Five Lemma, it follows \(\eta _{X}\) is an epimorphism. Hence, \(\eta _{X}\) is an isomorphism. Thus, \(\eta _{A}\) is also an isomorphism, and it means that \(\mathrm{Ext}_{R}^{1}(X,F)=0\). Now, by the dimension-shifting method, we have that \(\mathrm{Ext}_{R}^{i}(X,F)=0\) for all \(i>0\). \(\square \)

Corollary 2.11

If R is a GFC-ring, then every weak flat R-module is weak injective.

Proof

Let F be a weak flat R-module. Since R is a GFC-ring, R is self-weak injective. For any super finitely presented R-module N, it follows from [17, Proposition 3.1(1)] that \(\mathrm{Ext}_{R}^{i}(N,R)=0\) for all \(i\ge 1\). Hence, \(\mathrm{Ext}_{R}^{1}(N,F)=0\) by Proposition 2.10. So, F is weak injective. \(\square \)

Now, we can characterize the GFC-rings.

Theorem 2.12

The following statements are equivalent for a ring R:

  1. (1)

    R is a GFC-ring;

  2. (2)

    every super finitely presented R-module is Gorenstein projective;

  3. (3)

    \(G\text{- }\text{ s.gl. }\dim (R)=0\);

  4. (4)

    every super finitely presented R-module is Gorenstein flat;

  5. (5)

    \(G\text{- }\text{ s.gl. }\dim (R)<\infty \) and every weak flat R-module is weak injective;

  6. (6)

    \(G\text{- }\text{ s.gl. }\dim (R)<\infty \) and R is self-weak injective;

  7. (7)

    \(G\text{- }\text{ s.gl. }\dim (R)<\infty \) and every projective R-module is weak injective;

  8. (8)

    R is a GC-ring and every super finitely presented R-module can be embedded in a finitely generated free R-module.

Proof

We shall prove that \((4)\Leftrightarrow (3)\Leftrightarrow (2)\Leftrightarrow (1)\Rightarrow (5)\Rightarrow (6)\Rightarrow (7)\Rightarrow (2)\Rightarrow (8)\Rightarrow (1)\).

\((1)\Rightarrow (2)\) Let M be a super finitely presented R-module. Since R is a GC-ring, \(M^{*}\) is super finitely presented. Then, there exists a long exact sequence

$$\begin{aligned} \cdots \rightarrow F_{n}\rightarrow \cdots \rightarrow F_{1}\rightarrow F_{0}\rightarrow M^{*}\rightarrow 0, \end{aligned}$$

where each \(F_{i}\) is finitely generated projective. Set \(K_{i}=\mathrm{ker}(F_{i+1}\rightarrow F_{i})\) (\(i=0,1,\dots \)). Then, each \(K_{i}\) is super finitely presented. Since R is self-weak injective, \(\mathrm{Ext}_{R}^{1}(K_{i},R)=0\). Hence, we have the following long exact sequence

$$\begin{aligned} 0\rightarrow M^{**}\rightarrow F_{0}^{*}\rightarrow F_{1}^{*}\rightarrow \cdots \rightarrow F_{n}^{*}\rightarrow \cdots . \end{aligned}$$

Since M is reflexive by Theorem 2.8(1), we get the following long exact sequence

$$\begin{aligned} 0\rightarrow M\rightarrow F_{0}^{*}\rightarrow F_{1}^{*}\rightarrow \cdots \rightarrow F_{n}^{*}\rightarrow \cdots . \end{aligned}$$

Also, since M is super finitely presented, there exists the following long exact sequence

$$\begin{aligned} \cdots \rightarrow P_{n}\rightarrow \cdots \rightarrow P_{1}\rightarrow P_{0}\rightarrow M\rightarrow 0, \end{aligned}$$

where each \(P_{i}\) is finitely generated projective. Thus, we get the perfect revolution of M as follows:

$$\begin{aligned} \cdots \rightarrow P_{n}\rightarrow \cdots \rightarrow P_{1}\rightarrow P_{0}\rightarrow F_{0}^{*}\rightarrow F_{1}^{*}\cdots \rightarrow F_{n}^{*}\cdots , \end{aligned}$$

where \(M\cong \mathrm{ker}(P_{0}\rightarrow F_{0}^{*})\). Set L the cosyzygy of this sequence. Then, L is a super finitely presented R-module. Since R is self-weak injective, \(\mathrm{Ext}_{R}^{1}(L,R)=0\). It means that \(\mathrm{Hom}_{R}(-,Q)\) leaves the sequence exact whenever Q is a projective R-module. Therefore, M is Gorenstein projective.

\((2)\Rightarrow (1)\) If M is a super finitely presented R-module, then M is Gorenstein projective by the condition. It follows from [16, Corollary 2.4] that \(M^{*}\) is super finitely presented. Hence, R is a GC-ring. Also, since any super finitely presented R-module is Gorenstein projective by the condition, it follows from Lemma 2.9 that any super finitely presented R-module is reflexive. Hence, R is self-weak injective by Theorem 2.8(2).

\((2)\Leftrightarrow (3)\) Trivial.

\((2)\Leftrightarrow (4)\) This follows from [1, Theorem 3.3].

\((1)\Rightarrow (5)\) It follows from Corollary 2.11 that every weak flat R-module is weak injective. By (3)(\(\Leftrightarrow (1)\)), we have \(G\text{- }\text{ s.gl. }\dim (R)<\infty \).

\((5)\Rightarrow (6)\) It is clear since R is always weak flat.

\((6)\Rightarrow (7)\) Let P be a projective R-module. Then, there exists a free R-module F and a projective R-module Q such that \(P\bigoplus Q\cong F\). Since R is self-weak injective, F is weak injective by [17, Proposition 2.3(1)]. By [17, Proposition 2.3(1)] again, it follows that P is weak injective.

\((7)\Rightarrow (2)\) Let M be a super finitely presented R-module. Then, for any projective R-module P, \(\mathrm{Ext}_{R}^{1}(M,P)=0\) since P is weak injective by (7). It follows from [17, Proposition 3.1] and [18, Theorem 2.20] that M is Gorenstein projective.

\((2)\Rightarrow (8)\) If M is a super finitely presented R-module, then M is Gorenstein projective by (2). It follows from [16, Corollary 2.4] that \(M^{*}\) is super finitely presented. Hence, R is a GC-ring. Noting that M is finitely generated and Gorenstein projective, M can be always embedded to a finitely generated free module.

\((8)\Rightarrow (1)\) By the (8), we only need to prove that R is self-weak injective. Let M be a super finitely presented R-module. Then, there exists the exact sequence

$$\begin{aligned} F_{1}\rightarrow F_{0}\rightarrow M\rightarrow 0. \end{aligned}$$

Set \(N=\mathrm{cok}(F_{0}^{*}\rightarrow F_{1}^{*})\). Then, we get the following exact sequence,

$$\begin{aligned} 0\rightarrow M^{*}\rightarrow F_{1}^{*}\rightarrow F_{0}^{*}\rightarrow N \rightarrow 0. \end{aligned}$$

Hence, N is a super finitely presented R-module. By (8), N can be embedded to a finitely generated free module. So N is torsionless. On the other hand, from the short exact sequence \( F_{1}^{*}\rightarrow F_{0}^{*}\rightarrow N \rightarrow 0\), by Lemma 2.7, we have the following exact sequence

$$\begin{aligned} 0\rightarrow \mathrm{Ext}_{R}^{1}(M, R)\rightarrow N\rightarrow N^{**}\rightarrow \mathrm{Ext}_{R}^{2}(M, R)\rightarrow 0. \end{aligned}$$

Since N is torsionless, it means that \(\mathrm{Ext}_{R}^{1}(M, R)=0\). Therefore, R is self-weak injective. \(\square \)

By Corollary 2.5, it follows that every FC-ring is a GFC-ring. Next, we shall give an example to show that a GFC-ring is not necessarily an FC-ring. It is also an example of a GC-ring but not a coherent ring. We need the following lemma.

Lemma 2.13

Let R be ring. Then, the following statements hold:

  1. (1)

    \(G\text{- }\text{ s.gl. }\dim (R)\le \text{ s.gl. }\dim (R)\);

  2. (2)

    if R is a coherent ring, then \(\text{ w.gl. }\dim (R)=\text{ s.gl. }\dim (R)\).

Proof

(1) Clearly.

(2) See [17, Theorem 3.8]. \(\square \)

Example 2.14

A ring R is called a (n, d)-ring in [8] if every R-module has a finite n-presentation. As in [17, Remark 3.11], there exists a (2, 0)-ring R, but not a (0, 1)-ring (See [23, Theorem 3.4]), for which \(\text{ s.gl. }\dim (R)=0\) but \(\text{ w.gl. }\dim (R)\ne 0\). Since \(G\text{- }\text{ s.gl. }\dim (R)\le \text{ s.gl. }\dim (R)\) by Lemma 2.13 (1), it implies that \(G\text{- }\text{ s.gl. }\dim (R)=0\). On the other hand, R is not a coherent ring. Otherwise, \(\text{ s.gl. }\dim (R)=\text{ w.gl. }\dim (R)\) by Lemma 2.13 (2), and it means that \(\text{ w.gl. }\dim (R)=0\), which is a contradiction. Hence, R is a GFC-ring. Also, since R is not a coherent ring, R is not an FC-ring.

As in [12], a ring R is called an FGF-ring if every finitely generated R-module can be embedded in a free R-module. The so-called FGF-problem is whether an FGF-ring is a QF-ring. Comparing with this question, we consider a weaker case, i.e., whether R is a QF-ring if every finitely presented R-module can be embedded to a free module. We shall give an example to show that if every finitely presented R-module can be embedded in a free R-module, then R is not necessarily a QF-ring. By using Theorem 2.12, we start by a new characterization of an FC-ring.

Corollary 2.15

A ring R is a coherent ring, and every finitely presented R-module can be embedded in a free R-module if and only if R is an FC-ring.

Proof

(\(\Rightarrow \)) Since R is a coherent ring, R is a GC-ring by Corollary 2.4. Also, since every super finitely presented R-module is finitely presented, it means that every super finitely presented R-module can be embedded in a free R-module. Hence, R is a GFC-ring by Theorem 2.12. Since R is a coherent ring, it follows that R is a FC-ring by Corollary 2.6.

(\(\Leftarrow \)) Since R is an FC-ring, R is a coherent ring. Let M be a finitely presented R-module. Then, M is super finitely presented by Proposition 2.2. Since R is a GFC-ring, it follows from Theorem 2.12 that M can be embedded in a free R-module. \(\square \)

Lemma 2.16

R is a G-Dedekind domain if and only if R / (u) is a QF-ring for any nonzero nonunit \(u\in R\).

Proof

(\(\Rightarrow \)) This follows from [19, Theorem 2.11].

(\(\Leftarrow \)) If R / (u) is a QF-ring for any nonzero nonunit \(u\in R\), then R / (u) is an Artin ring by [29, Theorem 4.6.3]. Hence, for any nonzero proper ideal I of R, R / I is an Artin ring. It follows from [29, Theorem 4.3.19] that R is a Noetherian ring and \(\dim (R)=1\). Also, since every QF-ring is an FC-ring, it means that R / (u) is an FC-ring for any nonzero nonunit \(u\in R\). By [26, Theorem 4.2], it means that R is a G-Prüfer domain. By [26, Corollary 4.3], R is a G-Dedekind domain. \(\square \)

Example 2.17

Take any non-Noetherian Prüfer domain D. Then, D is a G-Prüfer domain, but not a G-Dedekind domain by [26, Corollary 4.3]. Hence, there exists a nonzero nonunit \(u\in D\) such that \(R=D/(u)\) is not a QF-ring by Lemma 2.16. But, since R is an FC-ring by [26, Theorem 4.2], it follows from Corollary 2.15 that every presented module R-module can be embedded a free module.

It follows from [13, Corollary 2.8] that if a ring R is a coherent ring, then R is an FGF-ring if and only if R is a QF-ring. Now, by Corollary 2.15, we can give a new approach to finish it.

Corollary 2.18

A ring R is a coherent FGF-ring if and only if R is a QF-ring.

Proof

(\(\Rightarrow \)) Suppose that R is an FGF-ring. We claim that R is a Noetherian ring. Let M be a finitely generated R-module. Then, M can be embedded in a free R-module F. Since R is a coherent ring, it means that F is a coherent R-module. Hence, M is finitely presented. And so R is a Noetherian ring. By Corollary 2.15, it follows that R is a Noetherian FC-ring. Let I be any ideal of R. Since R is a Noetherian ring, I is finitely generated. Hence, R / I is finitely presented. Since R is a FC-ring, R is self-FP-injective. It means that \(\mathrm{Ext}_{R}^{1}(R/I,R)=0\). It follows that R is self-injective. So, R is a QF-ring.

(\(\Leftarrow \)) Suppose that R is a QF-ring. Then, R is a coherent ring and every R-module is Gorenstein projective by [2, Proposition 2.6]. Hence, every finitely generated R-module can be always embedded a projective R-module. So, every finitely generated R-module can be always embedded a free R-module and R is an FGF-ring. Consequently, R is a coherent FGF-ring. \(\square \)

At last, we give the characterizations of a ring with \(G\text{- }\text{ s.gl. }\dim (R)\le 1\).

Theorem 2.19

The following statements are equivalent for a ring R:

  1. (1)

    every super finitely presented submodule of a projective R-module is Gorenstein projective;

  2. (2)

    \(G\text{- }\text{ s.gl. }\dim (R)\le 1\);

  3. (3)

    if M is super finitely presented R-module, \(\mathrm{Ext}_{R}^{1}(M,R)\) is finitely generated and if A is a super finitely presented torsionless R-module such that \(A^{*}\) is finitely generated, A is Gorenstein projective.

Proof

\((1)\Rightarrow (2)\) Let M be a super finitely presented R-module. Then, there exists a short exact sequence

$$\begin{aligned} 0\rightarrow N\rightarrow P\rightarrow M\rightarrow 0, \end{aligned}$$

where N is super finitely presented and P is finitely generated projective. Hence, N is Gorenstein projective by (1). So \(\text{ Gpd }_{R}M\le 1+\mathrm{Max}\{\text{ Gpd }_{R}N,\text{ Gpd }_{R}P\}=1\) by [29, Proposition 11.3.12 (2)]. It means that \(G\text{- }\text{ s.gl. }\dim (R)\le 1\).

\((2)\Rightarrow (1)\) Suppose that \(G\text{- }\text{ s.gl. }\dim (R)\le 1\). Let M be a super finitely presented submodule of a projective R-module P. Then, we have the short exact sequence \(0\rightarrow M\rightarrow P\rightarrow P/M\rightarrow 0\). Since M is super finitely presented, P / M is super finitely presented. Since \(G\text{- }\text{ s.gl. }\dim (R)\le 1\), it means that \(\text{ Gpd }_{R}(P/M)\le 1\). If \(\text{ Gpd }_{R}(P/M)=0\), then M is a Gorenstein projective module by [29, Proposition 11.1.10]. If \(\text{ Gpd }_{R}(P/M)= 1\), then \(\text{ Gpd }_{R}M=\text{ Gpd }_{R}(P/M)-1=0\) by [29, Proposition 11.3.12 (1)], whence M is Gorenstein projective.

\((1)\Rightarrow (3)\) Let M be a super finitely presented R-module. Then, there exists a short exact sequence

$$\begin{aligned} 0\rightarrow K\rightarrow P\rightarrow M\rightarrow 0, \end{aligned}$$

where K is super finitely presented and P is finitely projective. Hence, K is Gorenstein projective module by (1). Since P is projective, we have the following exact sequence

$$\begin{aligned} 0\rightarrow M^{*}\rightarrow P^{*}\rightarrow K^{*}\rightarrow \mathrm{Ext}_{R}^{1}(M,R)\rightarrow 0 \ \ (*). \end{aligned}$$

Noting that \(K^{*}\) is super finitely presented [16, Corollary 2.4], it implies that \(\mathrm{Ext}_{R}^{1}(M,R)\) is finitely generated. On other hand, let A be a super finitely presented torsionless R-module such that \(A^{*}\) is finitely generated. Then, there exists a finitely generated free R-module F such that \(F\rightarrow A^{*}\rightarrow 0\). Hence, we have \(0\rightarrow A^{**}\rightarrow F^{*}\). Since A is torsionless, \(A\rightarrow A^{**}\) is a monomorphism. Hence, A can be embedded in the free module \(F^{*}\). So, A as a submodule of the projective R-module \(F^{*}\) is a Gorenstein projective module by (1).

\((3)\Rightarrow (1)\) Let A be a super finitely presented submodule of a projective module P. Then, we have the short exact sequence \(0\rightarrow A\rightarrow P\rightarrow C\rightarrow 0\), where \(C=P/A\). Hence, A is torsionless and C is super finitely presented. So, \(\mathrm{Ext}_{R}^{1}(C,R)\) is finitely generated by (3). Also, from the following sequence,

$$\begin{aligned} 0\rightarrow C^{*}\rightarrow P^{*}\rightarrow A^{*}\rightarrow \mathrm{Ext}_{R}^{1}(C,R)\rightarrow 0, \end{aligned}$$

it follows that \(A^{*}\) is finitely generated. Hence, A is Gorenstein projective by the hypothesis. \(\square \)