# Some Characterizations ofw-Noetherian Rings and SM Rings

Authors: De Chuan Zhou; Hwankoo Kim; Xiaolei Zhang; Jin Xie
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this paper, we characterizew-Noetherian rings and SM rings. More precisely, in terms of the u-operation on a commutative ring R, we prove that R is w-Noetherian if and only if the direct limit of rGV-torsion-free injective R-modules is injective and that R is SM, which can be regarded as a regular w-Noetherian ring, if and only if the direct limit of GV-torsion-free (or rGV-torsion-free) reg-injective R-modules is reg-injective. As a by-product of the proof of the second statement, we also obtain that the direct and inverse limits of u-modules are both u-modules and that SM rings are regular w-coherent.


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## Body

## 1. Introduction

In this paper, we assume thatR is a commutative ring with identity with total quotient ring TR.Generalizing the properties of the integral domain to commutative rings makes it natural to consider regular ideals of rings, and many new kinds of rings that are defined by taking regular ideals emerge. Examples include the Krull ring that was introduced by Kennedy [1] and the Prüfer v-multiplication ring (PvMR) that was presented by Matsuda [2]. These can be referred to [3–5]. The w-operation can be used as a research tool for commutative rings, but it is inaccurate for the above kind of rings. For example, Wang and Kim introduced the concept of the w-semi-hereditary ring in [6], which is also a PvMD-like ring, where a connected ring R is w-semi-hereditary if and only if R is a Prüfer v-multiplication domain (PvMD). This tells us that a direct description of rings defined by regularity using the w-operation is still lacking. Recently, Zhang [7] introduced the regular w-flat module to provide some homological characterizations of the PvMR, which is a good attempt.In [8], Chang and Oh introduced the notion of general Krull rings, which are also defined by regularity. Also, they asked the following question. Is there a star operation ∗ on a ring so that a general Krull ring can be characterized as a ring in which each proper principal ideal can be written as a finite ∗-product of prime ideals [8, Question 0.2]? In order to answer the above question, they introduced a new star operation u on a ring R, and they showed that R is a general Krull ring if and only if each proper principal ideal of R is written as a finite u-product of prime ideals [8, Theorem 5.6].Coincidentally, theu-operation on a ring was also introduced by Tao [9], called the w∗-operation in her master thesis independently. Unlike Chang and Oh who studied the u-operation from a ring theory perspective, Tao studied this star operation from a module theory perspective.In more detail, Noetherian rings can be characterized by injectivity, which is known as the Cartan–Eilenberg–Bass theorem, i.e., a ringR is Noetherian if and only if the direct sum of any number of injective R-modules is injective, if and only if the direct sum of any countable number of injective R-modules is injective, if and only if every injective R-module is Σ-injective, if and only if the direct limit of injective R-modules is injective. An injective R-module E is said to be Σ-injective if every direct sum of copies of E is injective. In [10], Zhang et al. proved the w-theoretic analog of the Cartan–Eilenberg–Bass theorem for Noetherian rings. The w-Noetherian ring is defined to be the ring which satisfies the ascending chain condition (ACC) on w-ideals by Yin et al. [11]. Also, a ring R is w-Noetherian if and only if R satisfies the ACC on its u-ideals (see [8, Theorem 6.9] or [9, Proposition 4.1.9]).In [12], SM rings are introduced by Wang and Liao. A ring R is called SM if R satisfies the ACC on its regular w-ideals [12, Definition 5.10]. SM rings can be regarded as regular w-Noetherian ones loosely. Tao also proved that a ring R is SM if and only if R satisfies the ACC on its regular u-ideals [9, Theorem 4.2.1]. Thus, a natural question arises as to whether SM rings have both w-theoretic and u-theoretic analogs corresponding to the Cartan–Eilenberg–Bass theorem for Noetherian rings. Actually for this purpose, the notions of reg-injective and ∑-reg-injective R-modules are introduced in [12]. An R-module E is reg-injective if ExtR1R/I,E=0 for any regular ideal I of R [12, Definition 5.2]. A reg-injective R-module E is ∑-reg-injective if every direct sum of its copies is reg-injective [12, Definition 6.4]. In [12, Theorem 6.10] and [9, Theorem 4.2.5], it is proved that a ring R is SM if and only if the direct sum of any number of GV-torsion-free (or rGV-torsion-free) reg-injective R-modules is reg-injective, if and only if the direct sum of any countable number of GV-torsion-free (or rGV-torsion-free) reg-injective R-modules is reg-injective, if and only if every GV-torsion-free (or rGV-torsion-free) reg-injective R-module is Σ-reg-injective. However, what about the direct limit of GV-torsion-free (or rGV-torsion-free) reg-injective R-modules for SM rings? Motivated by this question, we first show that the u-operation can induce a torsion theory, denoted by τu, by a Gabriel topology(1)ℑ=I|Iis an ideal ofRwithIu=R.Also, forw-Noetherian rings, we complete the u-theoretic analog of the Cartan–Eilenberg–Bass theorem for Noetherian rings in terms of our existing knowledge of general torsion theory. Then, for SM rings, we complete both the w and u-theoretic analogs of the Cartan–Eilenberg–Bass theorem for Noetherian rings. In the process, the discussion of the direct limit of u-modules is also necessary.


## 2. Preliminaries

Now we introduce some notations and results needed in this paper from [9, 13]. Let J be a finitely generated ideal of R. If the natural homomorphism φ:R⟶J∗=HomRJ,R is an isomorphism, then J is called a GV-ideal, denoted by J∈GVR. Let M be an R-module. Define(2)torGVM=x∈M|Jx=0for someJ∈GVR.Thus,torGVM is a submodule of M. Also, M is said to be GV-torsion (resp., GV-torsion-free) if torGVM=M (resp., torGVM=0). Clearly R is a GV-torsion-free R-module [11, Corollary 1.5]. A GV-torsion-free module M is called a w-module if ExtR1R/J,M=0 for any J∈GVR. The w-envelope of a GV-torsion-free module M is the set given by(3)Mw=x∈EM|Jx⊆Mfor someJ∈GVR,where EM is the injective hull of M. It is easy to see that M is a w-module if and only if Mw=M. A nonzero ideal P of R is said to be a primew-ideal if P is both a prime ideal and a w-ideal and a maximalw-ideal if P is maximal in the set of all proper w-ideals of R. Note that each maximal w-ideal is prime [13, Theorem 6.2.14]. A GV-torsion-free module M is of finite type if Mw=Nw for some finitely generated submodule N of M [13, Proposition 6.4.2]. A sequence A⟶B⟶C of R-modules and homomorphisms is said to be w-exact if the sequence Am⟶Bm⟶Cm is exact for any maximal w-ideal m of R. An R-module M is said to be of finitely presented type if there is a w-exact sequence F1⟶F0⟶M⟶0, where F1 and F0 are finitely generated and free modules [13, Definition 6.4.9].An idealI of R is regular if I contains a nonzero divisor. An ideal J of R is called anrGV-ideal if J is a regular GV-ideal. Let rGVR denote the set of all rGV-ideals of R. Then, rGVR is a multiplicative system of ideals of R, i.e., rGVR satisfies that (i) R∈rGVR and (ii) if J1,J2∈rGVR; then, J1J2∈rGVR. It is clear that rGVR⊆GVR. But the converse does not hold in general.Example 1.
(see [9, Example 2.1.1]). Let F be a field, and let D=Fy,z and K=Fy,z, where y and z are indeterminates over F. Then, for the trivial extension R=D∝K/Dw, we have that TR=R. Then, rGVR=R. By [14, Theorem 4.7], R is not a DW ring, i.e., the ring satisfies that every ideal of R is a w-ideal. Then, GVR≠R by [13, Theorem 6.3.12]. Thus, GVR⊈rGVR.
LetM be an R-module. Define(4)torrGVM=x∈M|Jx=0for someJ∈rGVR.
Thus,torrGVM is a submodule of M. Also, M is said to be rGV-torsion (resp., rGV-torsion-free) if torrGVM=M (resp., torrGVM=0). It is clear that any GV-torsion-free R-module is rGV-torsion-free, while any rGV-torsion R-module is GV-torsion.Example 2.
(see [9, Example 2.2.4]). LetJ∈GVR, butJ∉rGVR. Then, R/JisGV-torsion, but notrGV-torsion.Proposition 1 (see [9, Proposition 2.2.6 and Proposition 2.2.7]).
(1)
AnR-moduleMisrGV-torsion if and only ifHomRM,N=0for anyrGV-torsion-freeR-moduleN.(2)
AnR-moduleNisrGV-torsion-free if and only ifHomRM,N=0for anyrGV-torsionR-moduleM.(3)
LetMibe a family ofR-modules. Then, ∏MiisrGV-torsion-free if and only if eachMiisrGV-torsion-free.(4)
IfMis anrGV-torsion-freeR-module, thenEMis alsorGV-torsion-free.AnrGVR-torsion-free R-module M is called au-module if ExtR1R/J,M=0 for any J∈rGVR[9, Definition 3.1.1]. In [9, Definition 3.2.1], the u-envelope of an rGV-torsion-free R-module M is the set given by(5)Mu=x∈EM|Jx⊆Mfor someJ∈rGVR.It is clear thatMuu=Mu and that J∈rGVR if and only if Ju=R[9, Proposition 3.2.5].Proposition 2 (see [9, Theorem 3.1.7 and Theorem 3.2.12]).
The following statements are equivalent for anrGV-torsion-freeR-module.(1)
Mis au-module.(2)
Mu=M.(3)
If0⟶M⟶N⟶C⟶0is an exact sequence in whichNis au-module, thenCisrGV-torsion-free.(4)
There exists an exact sequence0⟶M⟶N⟶C⟶0such thatNis au-module andCisrGV-torsion-free.(5)
ExtR1C,M=0for anyrGV-torsionR-moduleC.(6)
ExtR1Au/A,M=0for anyrGV-torsionR-moduleA.An idealI of R is called a u-ideal if I is a u-module. It is clear that w-ideals of R are u-ideals, but the converse does not hold. One can refer to [8, Example 4.9]. A nonzero ideal p of R is said to be a prime u-ideal if p is both a prime ideal and a u-ideal, denoted by p∈u-SpecR, and a maximal u-ideal if p is maximal in the set of all proper u-ideals of R, denoted by p∈u-MaxR. Note that each maximal u-ideal is prime [9, Theorem 3.3.4].Proposition 3.
(see [9, Theorem 3.3.5, Theorem 3.3.6, and Theorem 3.3.7])(1)
AnR-moduleMisrGV-torsion if and only ifMm=0for any maximalu-idealm.(2)
LetMbe anrGV-torsion-freeR-module. Then, Mp=Mup.(3)
LetMbe anrGV-torsion-freeR-module and letAandBbe submodules ofM. Then, Au=Buif and only ifAm=Bmfor any maximalu-idealmofR.AnR-module M is said to be u-finitely generated if there exists some finitely generated submodule N of M such that M/N is an rGV-torsion R-module [9, Definition 3.4.1]. An rGV-torsion-free R-module M is u-finitely generated if and only if Mu=Bu for some finitely generated submodule B of M[9, Proposition 3.4.3].Proposition 4 (see [9, Proposition 3.4.5]).
Let0⟶A⟶B⟶C⟶0be an exact sequence ofR-modules.(1)
IfAandCareu-finitely generated, thenBisu-finitely generated.(2)
IfBisu-finitely generated, thenCisu-finitely generated.


## 3. Injective Modules overw-Noetherian Rings

Next, as in [15], we show that the u-operation can induce a torsion theory, denoted by τu, by a Gabriel topology(6)ℑ=I∣Iis an ideal ofRwithIu=R.By the proof of [16, Proposition 4.6], the class of all τu-torsion R-modules, denoted by TτuR, is the set {M|M is an R-module and 0:Rmu=R for each nonzero element m∈M}. Let TrGVR denote the set of all rGV-torsion R-modules. The following proposition shows that TrGVR and TτuR coincide. Thus, τu-torsion-free (resp., τu-torsion) modules and rGV-torsion-free (resp., rGV-torsion) modules coincide. The proof of the following proposition is very similar to that of [15, Proposition 2.10]; however, we give a proof for completeness.Proposition 5.
For a ringR, TrGVR=TτuR.Proof.
Note thatM is an rGV-torsion R-module if and only if for any nonzero element m∈M there exists some J∈rGVR such that mJ=0; if and only if 0:Rmu=R for any nonzero element m∈M; and if and only if M is a τu-torsion R-module.
Now we recall some terminology in [17], which is similar to that in [1515]. Let M be an R-module. A submodule N of M is called τu-pure (resp., τu-dense) in M if M/N is rGV-torsion-free (resp., rGV-torsion). Obviously if N is a τu-dense submodule of an rGV-torsion-free R-module M, then Nu=Mu. Set CτuMN:=x∈M|N:Rxu=R, which is called the τu-closure of N in M. Then, N is called τu-closed in M if CτuMN=N. It is easy to verify that if M is rGV-torsion-free, then CτuMN=Nu∩M; N is τu-dense in M if and only if CτuMN=M; and τu-closed submodules of M and its τu-pure submodules coincide.Lemma 1.
LetM be an rGV-torsion-free R-module. If M is a u-module, then τu-pure submodules and u-submodules of M coincide.Proof.
LetN be a submodule of M. Then, the sequence(7)HomRRJ,MN⟶ExtR1RJ,N⟶ExtR1RJ,M,is exact for any J∈rGVR. Note that if N is a τu-pure submodule of M, then M/N is rGV-torsion-free. So, HomRR/J,M/N=0. Obviously, ExtR1R/J,M=0 because M is a u-module. Thus, ExtR1R/J,N  = 0, and so N is a u-module. Conversely, if N is a u-module, then it is easy to prove that M/N is rGV-torsion-free. Thus, N is a τu-pure submodule of M.
Recall that anR-module M is said to be τu-Noetherian if M satisfies the ACC on its τu-pure submodules [16, p. 175]. Thus, an rGV-torsion-free R-module M is τu-Noetherian if M satisfies the ACC on its u-submodules by Lemma 1. Thus, an rGV-torsion-free τu-Noetherian R-module is also called u-Noetherian in [9]. A ring R is τu-Noetherian if R is a τu-Noetherian R-module. Note that R is rGV-torsion-free over R. Then, R is τu-Noetherian if and only if R satisfies ACC on its u-ideals.
In [9] or [8], it is proved that τu-Noetherian rings coincide with w-Noetherian ones. Also, in terms of u-operations, some characterizations of w-Noetherian rings are provided in [9].Proposition 6 (see [9, Proposition 4.1.9 and Theorem 4.1.10]).
The following statements are equivalent for a ringR.(1)
Ris aw-Noetherian ring.(2)
Risτu-Noetherian.(3)
Every ideal isu-finitely generated, i.e., for each idealIofR, there exists some finitely generated subidealI0ofIsuch thatIu=I0u.(4)
Everyu-ideal isu-finitely generated.(5)
Every nonempty set ofu-ideals ofRhas a maximal element.(6)
Every primeu-ideal ofRisu-finitely generated overR.(7)
The direct sum of any number ofrGV-torsion-free injectiveR-modules is injective.(8)
The direct sum of any countable number ofrGV-torsion-free injectiveR-modules is injective.(9)
EveryrGV-torsion-free injectiveR-module is∑-injective.Remark.
(1)
In fact, with the help of the language of torsion theory, the equivalences of (2)–(6) of Proposition6 can be obtained directly by [16, Proposition 20.1] or [17, Proposition 2.3.3], while the equivalences of (2) and (7)–(9) of Proposition 6 can be obtained directly by [16, Proposition 20.17].(2)
Althoughτu-Noetherian rings coincide with w-Noetherian ones, w-Noetherian R-modules are not necessarily τu-Noetherian. An R-module M is w-Noetherian if M satisfies ACC on its w-submodules. If N is a w-submodule of M, then N is GV-torsion-free, which implies that N is rGV-torsion-free. Since Nw=N, we can get that Nu=N. Then, N is a τu-pure submodule of M. So, τu-Noetherian modules are w-Noetherian by their definitions. But the converse does not hold by [9, Example 4.1.7]. In more detail, let J∈GVR and J∉rGVR. Then, R/J is GV-torsion, not rGV-torsion over R. Set LR/J:=α∈R/J|ifJα=0whereJ∈rGVR,thenα=0. Thus, LR/J is an rGV-torsion-free submodule of R/J and L:=LR/Ju∩R/J is a τu-pure submodule of R/J. Let M be a direct sum of countably infinite number of R/J. Then, M is a w-Noetherian R-module since M is GV-torsion. Note that the chain L⊆L⊕L⊆L⊕L⊕L⊆… of τu-pure submodules of M is not stationary. Then, M is not a τu-Noetherian R-module.
Next, with the help of the language of torsion theory, we can get more characterizations ofw-Noetherian rings in terms of u-operations. For this purpose, first we need some notions. An R-module M is τu-finitely generated if M has a finitely generated τu-dense submodule [16, p. 157]. Thus, an rGV-torsion-free R-module M is τu-finitely generated (also u-finitely generated in this case) if there exists a finitely generated submodule N of M such that Nu∩M=M, equivalently Nu=Mu. An R-module M is τu-finitely presented if it is isomorphic to F/K, where F is a finitely generated free R-module and K is a τu-finitely generated submodule of F[16, p. 164]. A ring R is τu-coherent if every finitely generated ideal of R is τu-finitely presented [18, Definition 1.2]. From [18, Theorem 3.3], we can get that R is a τu-coherent ring if and only if the direct limit of rGV-torsion-free FP-injective R-modules is FP-injective. Recall that an R-module M is said to be FP-injective if ExtR1N,M=0 for all finitely presented R-modules N. It can be also called an absolutely pureR-module. For more details, one can refer to [16].
Now, based on Proposition6, we complete the u-version of the Cartan–Eilenberg–Bass theorem for Noetherian rings. For this, we need the following.Lemma 2.
LetMi|i∈Γbe a family of rGV-torsion-freeR-modules. Then, lim⟶Miandlim←Miare alsorGV-torsion-free.Proof.
Let0⟶J⟶R⟶R/J⟶0 be an exact sequence, where J∈rGVR. Then, R/J is finitely presented. Thus, HomRR/J,lim⟶Mi≅lim⟶HomRR/J,Mi for any J∈rGVR by [10, Lemma 2.1]. So, lim⟶Mi is rGV-torsion-free. Since lim←Mi is a submodule of ∏i∈ΓMi, which is rGV-torsion-free by Proposition 1 (3), it is clear that lim←Mi is rGV-torsion-free.Theorem 1.
The following statements are equivalent for a ringR.(1)
Ris aw-Noetherian ring.(2)
Every finitely generatedR-module isτu-finitely presented.(3)
Every finitely generatedR-module isτu-Noetherian.(4)
The direct limit ofrGV-torsion-free injectiveR-modules is injective.(5)
rGV-torsion-free FP-injectiveR-modules andrGV-torsion-free injective ones coincide.Proof.
1⇒2 Let M be a finitely generated R-module. Then, there exists an exact sequence 0⟶K⟶F⟶M⟶0, where F is a finitely generated free R-module. Since w-Noetherian rings and τu-Noetherian ones coincide, it follows that F is a τu-Noetherian R-module by [16, Proposition 20.4]. Then, K is τu-finitely generated [16, Proposition 20.1]. Thus, M is τu-finitely presented by [16, Proposition 19.3].2⇒1 Let I be an ideal of R. Since R/I is finitely generated over R, it is τu-finitely presented by (2). Then, I is τu-finitely generated by [16, Proposition 19.3]. Thus, R is a τu-Noetherian ring again by [16, Proposition 20.1] or [17, Proposition 2.3.3]. Then, R is w-Noetherian.1⇒3 Let M be a finitely generated R-module. Then, there exists an exact sequence 0⟶K⟶F⟶M⟶0, where F is a finitely generated free R-module. By [16, Proposition 20.4], F is a τu-Noetherian R-module. Again by [16, Proposition 20.4], M is also a τu-Noetherian R-module.3⇒1 It is clear.1⇒5 The proof of this implication is very similar to that of 1⇒6 in [15, Theorem 3.17]; however, we give a proof for completeness. Assume that R is a w-Noetherian ring. For any ideal I of R, I is u-finitely generated over R by Proposition 6. Thus, there exists a finitely generated subideal I1 of I such that Iu=I1u. So, I/I1 is an rGV-torsion R-module. Let M be an rGV-torsion-free FP-injective R-module. Then, for the exact sequence 0⟶I/I1⟶R/I1⟶R/I⟶0, we can get the following exactsequence:(8)HomRI/I1,M⟶ExtR1RI,M⟶ExtR1RI1,M.SinceI/I1 is rGV-torsion over R and M is rGV-torsion-free over R, it follows that HomRI/I1,M=0 by Proposition 1 (1). Also, since M is FP-injective, we can get that ExtR1R/I1,M=0. Then, ExtR1R/I,M=0. Thus, M is an injective R-module.5⇒1Note that the direct sum of FP-injective R-modules is FP-injective by [19, p. 564]. Then, (1) holds by the equivalence of (1) and (7) in Proposition 6.2+5⇒4 By (2), R is τu-coherent. Then, we can get that the direct limit of rGV-torsion-free FP-injective R-modules is FP-injective by [4, Theorem 3.3]. By (6) and Lemma 2, the direct limit of rGV-torsion-free injective R-modules is injective.4⇒1Since GV-torsion-free R-modules are rGV-torsion-free, we can get that the direct limit of GV-torsion-free injective modules is injective by (4). Then, R is w-Noetherian by [10, Theorem 2.9].


## 4. The Direct and Inverse Limits ofu-Modules

One main purpose of this paper is to generalize the Cartan–Eilenberg–Bass theorem for Noetherian rings to SM rings. For this, the discussion of the direct limit ofu-modules is necessary [20]. First we show that any rGV-ideal of R is τu-finitely presented.LetM be an R-module, M∗=HomRM,R, and EndRM=HomRM,M. Then, we have the natural homomorphism η:M⊗RM∗⟶EndRM by(9)ηx⊗fy=fyx,x,y∈M,f∈M∗.Lemma 3 (see [13, Theorem 2.6.17]).
LetMbe anR-module. Then, Mis finitely generated projective if and only ifηis an isomorphism.Lemma 4.
LetMbe anrGV-torsion-freeR-module and letηbe as in the above. Ifηmis an isomorphism for anym∈u-MaxR, thenMisu-finitely generated.Proof.
SinceM is rGV-torsion-free, it is easy to see that EndRM is also rGV-torsion-free. Note that Imηm=EndRMm by considering that ηm is an isomorphism for any m∈u-MaxR. Then, Imηu=EndRMu by Proposition 3. Thus, there exists some J∈rGVR such that J1M⊆Imf, where 1M denotes the identity map on M. Set J:=a1,a2,…,an. Then, for any i=1,2,…,n, there are finite sets xi1,xi2,…,xiti⊆M and fi1,fi2,…,fiti⊆M∗ such that ai1M=η∑j=1tixij⊗fij. Let B be the submodule of M generated by xij|i=1,2,…,n;j=1,2,…,ti. Then, for any x∈M, we have aix=ai1Mx=η∑j=1tixij⊗fijx=∑j=1tifijxxij∈B. Thus, JM⊆B, which implies that M⊆Bu. It is clear that Bu⊆Mu. Then, Mu=Bu. Therefore, M is u-finitely generated.Lemma 5.
LetJ∈rGVR. Then, ExtR1J,NisrGV-torsion for anyR-moduleN.Proof.
LetJ∈rGVR. For the exact sequence 0⟶J⟶R⟶R/J⟶0, we can get an exact sequence(10)0=ExtR1R,N⟶ExtR1J,N⟶ExtR2RJ,N,where N is an R-module. Note that ExtR2R/J,N is an R/J-module and so an rGV-torsion R-module. Then, ExtR1J,N is rGV-torsion over R.
LetSbe a multiplicatively closed set of R. An R-module M is said to be S-torsion if MS=0, and that M is said to be S-torsion-free if sx=0, for s∈S and x∈M, implies x=0.Lemma 6.
LetJ∈rGVRandm∈u-MaxR. Then, HomRJ,Nm≅HommJm,Nmfor anyR-moduleN.Proof.
SetS≔R\m and set C≔x∈N|sx=0for somes∈S. Then, C is S-torsion and N/C is S-torsion-free. For the exact sequence 0⟶C⟶N⟶N/C⟶0, we can get the following exact sequence:(11)0⟶HomRJ,C⟶HomRJ,N⟶HomRJ,NC⟶ExtR1J,C.
By Lemma1, ExtR1J,Cm=0. Note that HomRJ,C is S-torsion. Then, HomRJ,Cm=0. Thus, HomRJ,Nm≅HomRJ,N/Cm. By [21, Lemma 1.7], HomRJ,N/Cm≅HomRmJm,N/Cm. Note that N/Cm=Nm/Cm=Nm. Then, HomRJ,Nm≅HomRmJm,Nm.Theorem 2.
AnyrGV-ideal of R is τu-finitely presented.Proof.
LetJ∈rGVR. Then, there exits an exact sequence 0⟶A⟶F⟶J⟶0, whereF is finitely generated free R module. Let m be a maximal u-ideal of R. Then, ExtR1J,Nm=0 for any R-module N by Lemma 1. Note that Jm=Rm. Then, ExtRm1Jm,Nm=0. Thus, we have the following commutative diagram (12) with exact rows:(12)
Note thatf1 is an isomorphism by Lemma 5 and f2 is an isomorphism by [13, Theorem 2.6.16 (1)]. Then, HomRA,Nm≅HomRmAm,Nm. In particular, A∗m≅Am∗ and EndRAm≅EndRmAm. For the exact sequence 0⟶Am⟶Fm⟶Jm⟶0, since Fm and Jm are finitely generated and free over Rm, we can get that Am is also finitely generated free over Rm.
Consider the following commutative diagram:(13)
By Lemma3, the arrow in the bottom row is an isomorphism. Note that the vertical arrows are isomorphisms by the above. Then, the arrow in the top row is an isomorphism. So, η:A⊗RA∗⟶EndRA is a u-isomorphism. Note that A is an rGV-torsion-free R-module. Then, A is u-finitely generated over R by Lemma 4. Thus, J is τu-finitely presented over R.
Next, we show that the direct and inverse limits ofu-modules are also u-modules. To do so, we need the following.Lemma 7.
LetMi be a family of rGV-torsion-free R-modules. Then, lim⟶HomRJ,Mi≅HomRJ,lim⟶Mifor eachJ∈rGVR.Proof.
LetJ∈rGVR. Since J is finitely generated, there is a finitely generated free module F such that 0⟶K⟶F⟶J⟶0 is an exact sequence of R-modules. Note that J is τu-finitely presented by Theorem 2-, and so K is u-finitely generated. Since K is rGV-torsion-free, we can get that there exists a finitely generated submodule K1 of K such that Ku=K1u. Then, there is an exact sequence F1⟶fK1⟶0, where F1 is a finitely generated free R-module. Thus, we can get an exact sequence F1⟶fK⟶K/Imf⟶0. For any maximal u-ideal m, K/Imfm=Km/Imfm=K1m/Imfm=0. So, K/Imf is rGV-torsion. Then, we have the following commutative diagram (14) with exact rows:(14)
By Lemma2, lim⟶Mi is rGV-torsion-free. Then, it follows from Proposition 1 (1) that(15)HomRKImf,lim⟶Mi=0,HomRKImf,Mi=0.
By [10, Lemma 2.1], γ is an isomorphism. Then, β is a monomorphism by the Five Lemma.
Consider the following commutative diagram (16):(16)
By the above proof,β is a monomorphism. Again by [10, Lemma 2.1], f2 is an isomorphism. Hence, f1 is an epimorphism by the Five Lemma and f1 is a monomorphism also by the above proof. Therefore, f1 is an isomorphism.Corollary 1.
LetMi be a family of rGV-torsion-free R-modules. If K is rGV-torsion-free and u-finitely generated, then(17)β:lim⟶HomRK,Mi≅HomRK,lim⟶Mi,is a monomorphism.Theorem 3.
LetMibe a family ofu-modules overR. Then, lim⟶Miis au-module.Proof.
LetJ∈rGVR. Then, 0⟶J⟶R⟶R/J⟶0 is an exact sequence. Consider the following commutative diagram:(18)
By Lemma7, f1 is an isomorphism. Then, f2 is an isomorphism by the Five Lemma. Since each Mi is a u-module, ExtR1R/J,lim⟶Mi≅lim⟶ExtR1R/J,Mi=0. Hence, lim⟶Mi is a u-module.Lemma 8 (see [22, Theorem 2.22]).
The inverse limit of an inverse systemMi,ψijof modules exists andlim←Mi=ai∈∏Miai=ψijaj,wheneveri≤j.Theorem 4.
LetMi,ψijbe an inverse system ofu-modules overR. Then, lim←Miis au-module.Proof.
It suffices to show that everyR-homomorphism f:J⟶lim←Mi can be extended to R for each J∈rGVR.
Letλ:lim←Mi⟶∏Mi be an embedding and let λ1:J⟶R be an embedding. Consider the following diagram:(19)
Since∏Mi is a u-module, there is a homomorphism g:R⟶∏Mi such that gλ1=λf. Set g1:=ai∈∏Mi. For each x∈J, fx=gx=x⋅ai=xai∈lim←Mi. Then, xai=ψijxaj=xψijaj whenever i≤j by Lemma 8. Hence, Jai−ψijaj=0. Because Mi is rGV-torsion-free, we can get that ai−ψijaj=0, which implies ai=ψijaj whenever i≤j. Thus, g1=ai∈lim←Mi again by Lemma 8. Therefore, lim←Mi is a u-module.


## 5. Reg-Injective Modules over SM Rings

In this section, for SM rings, we complete both thew and the u-theoretic analogs of the Cartan–Eilenberg–Bass theorem for Noetherian rings. In [9], some characterizations are given in terms of u-operations.Proposition 7 (see [9, Theorem 4.2.1]).
The following statements are equivalent for a ringR.(1)
Ris an SM ring.(2)
Rsatisfies ACC on its regularu-ideals.(3)
Every regularu-ideal ofRisu-finitely generated.(4)
Every nonempty set of regularu-ideals ofRhas a maximal element.(5)
Every regular primeu-ideal ofRisu-finitely generated.(6)
Every regular ideal ofRisu-finitely generated.Theorem 5.
LetRbe an SM ring. Then, every finitely generated regular ideal ofRisτu-finitely presented.Proof.
LetI=a0,a1,…,an be a finitely generated regular ideal of R, where a0 is a regular element in R. We prove this by induction on n. If n=0, then I≅R, which implies that I is τu-finitely presented. Now assume that Ik=a0,a1,…,ak is τu-finitely presented, where k<n. Then, we have the following commutative diagram (20) with exact rows:(20)where f1r0,r1,…,rk=r0a0+r1a1+…+rkak for any r0,r1,…,rk∈Rk+1 and f2r0,r1,…,rk+1=r0a0+r1a1+…+rk+1ak+1for anyr0,r1,…,rk+1∈Rk+2.
Now we explain whyIk:RRak+1≅Lk+1/Lk. Define g:Lk+1⟶Ik:RRak+1 by gr0,r1,…,rk+1=rk+1 for any r0,r1,…,rk+1∈Lk+1. Then, it is easy to verify that g is well defined. For any r∈Ik:RRak+1, rak+1∈Ik. Then, there exists some elements r0,r1,…,rk such that rak+1=r0a1+r1a1+…+rkak. Thus, −r0,−r1,…,−rk,r∈Lk+1 and g−r0,−r1,…,−rk,r=r, which implies that g is an epimorphism. It is clear that Lk⊆Kerg. For any r0,r1,…,rk+1∈Kerg, gr0,r1,…,rk+1=rk+1=0. Then, r0,r1,…,rk∈Lk. Then, Kerf⊆Lk. Thus, Kerf=Lk. So,Ik:RRak+1≅Lk+1/Lk.
Note thatIk⊆Ik:RRak+1. Then, Ik:RRak+1 is a regular ideal of R. Since R is an SM ring, we can get that Ik:RRak+1 is u-finitely generated by Proposition 7. By assumption, Ik is τu-finitely presented, and so Lk is u-finitely generated [3, Proposition 19.3]. Thus, Lk+1 is u-finitely generated by Proposition 4. Hence, Ik+1 is τu-finitely presented.
It is well known that Noetherian rings are coherent ones, andw-Noetherian rings are w-coherent ones [13, p. 393]. Recall that R is w-coherent if each finite type ideal of R is of finitely presented type [13, Definition 6.9.14]. By the same way of Theorem 5, we can get that if R is an SM ring, then every finitely generated regular ideal of R is of finitely presented type, where such a ring is called regularw-coherent [7, Definition 2.4]. Then, we can get the following corollary, which is corresponding to the classical result.Corollary 2.
SM rings are regularw-coherent.Next, our purpose is to complete the Cartan–Eilenberg–Bass theorem for SM rings. To do so, we need the following.Lemma 9.
LetMibe a family ofrGV-torsion-freeR-modules. IfRis an SM ring, thenlim⟶HomRI,Mi≅HomRI,lim⟶Mifor any finitely generated regular idealIofR.Proof.
LetI be a finitely generated regular ideal of R. Since R is an SM ring, we can get that I is τu-finitely generated by Theorem 5. Then, there exists an exact sequence 0⟶K⟶F⟶I⟶0, where Fis finitely generated and free over R and K is u-finitely generated. Thus, we have the following commutative diagram (21) with exact rows:(21)
By Corollary1, f3 is a monomorphism. By [10, Lemma 2.1], f2 is an isomorphism. Then, f1 is an epimorphism by the Five Lemma. Note that f1 is a monomorphism again by Corollary 1. Thus, f1 is an isomorphism.Lemma 10.
LetNbe anrGV-torsion-freeR-module and letMbe au-module. Then, HomRN,M≅HomRNu,M.Proof.
For the exact sequence0⟶N⟶Nu⟶Nu/N⟶0, we can get an exact sequence 0⟶HomRNu/N,M⟶HomRNu,M⟶HomRN,M⟶ExtRNu/N,M. Note thatNu/N is rGV-torsion and M is a u-module. Then, HomRNu/N,M=ExtRNu/N,M=0 by Proposition 1 (1) and Proposition 2. Thus, HomRN,M≅HomRNu,M.Proposition 8.
LetMibe a family ofu-modules. IfRis an SM ring, thenlim⟶ExtR1R/I,Mi≅ExtR1R/I,lim⟶Mifor any regularu-idealIofR.Proof.
LetI be a regular u-ideal of R. From the exact sequence 0⟶I⟶R⟶R/I⟶0, we have the following commutative diagram (22) with exact rows:(22)
By [10, Lemma 2.1], f1 is an isomorphism. Since R is SM, we can get that I is u-finitely generated by Proposition 7. Then, there exists a finitely generated subideal I0 of I such that I=I0u. It is easy to verify that I0 is regular. Thus, lim⟶HomRI0u,Mi≅lim⟶HomRI0,Mi by Lemma 10. lim⟶HomRI0,Mi≅HomRI0,lim⟶Mi by Lemma 9. HomRI0,lim⟶Mi≅HomRI0u,lim⟶Mi by Theorem 3 and Lemma 10. Then, lim⟶HomRI,Mi≅HomRI,lim⟶Mi.
Recall that anR-module E is reg-injective if ExtR1R/I,E=0 for any regular ideal I of R [12, Definition 5.2].Proposition 9.
LetRbe a commutative ring and letEbe au-module overR. Then, the following statements are equivalent.(1)
E is a reg-injective R-module.(2)
ExtR1R/I,E=0 for any regular u-ideal I of R.(3)
For any regularu-ideal I of R, every homomorphism f:I⟶E can be extended to R.Proof.
1⇒2 This is trivial.2⇒1 Let I be a regular ideal of R. Then, Iu is a regular ideal of R and Iu/I is rGV-torsion. Since E is a u-module, we can get that ExtR1Iu/I,E=0 by Proposition 2. From the exact sequence 0⟶Iu/I⟶R/I⟶R/Iu⟶0, we can get an exact sequence(23)0=ExtR1R/Iu,E⟶ExtR1R/I,E⟶ExtR1Iu/I,E=0.Thus,ExtR1R/I,E=0. Therefore, E is injective.2⇔3 This is clear.Recall that a reg-injectiveR-module E is ∑-reg-injective if any direct sum of its copies is reg-injective [12, Definition 6.4].Theorem 6.
The following statements are equivalent for a ringR.(1)
R is an SM ring.(2)
The direct sum of any number ofrGV-torsion-free reg-injective R-modules is reg-injective.(3)
The direct sum of any number ofGV-torsion-free reg-injective R-modules is reg-injective.(4)
The direct sum of any countable number ofrGV-torsion-free reg-injective R-modules is reg-injective.(5)
The direct sum of any countable number ofGV-torsion-free reg-injective R-modules is reg-injective.(6)
EveryrGV-torsion-free reg-injective R-module is ∑-reg-injective.(7)
EveryGV-torsion-free reg-injective R-module is ∑-reg-injective.(8)
The direct limit ofrGV-torsion-free reg-injective R-modules is reg-injective.(9)
The direct limit ofGV-torsion-free reg-injective R-modules is reg-injective.Proof.
1⇔2⇔4⇔6 See [9, Theorem 4.1.10].1⇔3⇔5⇔7 See [12, Theorem 6.10].1⇒8 Let Ei be a family of rGV-torsion-free reg-injective R-modules. Then, each Ei is a u-module by definition. Since R is an SM ring, we can get that ExtR1R/I,lim⟶Ei≅lim⟶ExtR1R/I,Ei=0 for any regular u-ideal I of R by Proposition 8. Note that lim⟶Ei is a u-module by Theorem 3. Then, lim⟶Ei is reg-injective by Proposition 9.8⇒9 This follows by the fact that GV-torsion-free R-modules are rGV-torsion-free.9⇒5 This follows by the fact that a direct sum is a direct limit of finite sums.



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