Some Characterizations of <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2724395pt" id="M1" height="8.14084pt" version="1.1" viewBox="-0.0657574 -7.8684 12.3952 8.14084" width="12.3952pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-120" d="M689 332C689 394 670 448 646 448C620 448 597 421 597 396C597 386 600 381 608 372C619 359 620 334 620 315C620 150 538 45 454 45C414 45 386 67 386 122C386 138 388 158 394 180L457 426L452 432L377 416L315 156C302 100 259 45 216 45C176 45 148 67 148 122C148 133 152 158 156 180C162 212 173 259 194 332C201 357 206 384 206 405C206 430 198 448 174 448C125 448 66 406 23 342L43 319C84 368 110 383 121 383C126 383 128 382 128 377C128 370 127 359 122 343C99 268 84 204 77 156C74 137 70 111 70 104C70 25 125 -12 180 -12C228 -12 276 12 319 50C338 8 378 -12 418 -12C549 -12 689 166 689 332Z"/></g></svg>-</span>Noetherian Rings and SM Rings

Zhou, De Chuan; Kim, Hwankoo; Zhang, Xiaolei; Xie, Jin

doi:https://doi.org/10.1155/2022/7403502

Journal of Mathematics

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Research Article | Open Access

Volume 2022 | Article ID 7403502 | https://doi.org/10.1155/2022/7403502

Some Characterizations of -Noetherian Rings and SM Rings

De Chuan Zhou,¹Hwankoo Kim,²Xiaolei Zhang,³and Jin Xie⁴

Academic Editor: Marco Fontana

Received12 Nov 2022

Revised28 Nov 2022

Accepted29 Nov 2022

Published30 Dec 2022

Abstract

In this paper, we characterize -Noetherian rings and SM rings. More precisely, in terms of the -operation on a commutative ring , we prove that is -Noetherian if and only if the direct limit of -torsion-free injective -modules is injective and that is SM, which can be regarded as a regular -Noetherian ring, if and only if the direct limit of -torsion-free (or -torsion-free) reg-injective -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 -modules are both -modules and that SM rings are regular -coherent.

1. Introduction

In this paper, we assume that is a commutative ring with identity with total quotient ring .

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 -multiplication ring (PvMR) that was presented by Matsuda [2]. These can be referred to [3–5]. The -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 -semi-hereditary ring in [6], which is also a PvMD-like ring, where a connected ring is -semi-hereditary if and only if is a Prüfer -multiplication domain (PvMD). This tells us that a direct description of rings defined by regularity using the -operation is still lacking. Recently, Zhang [7] introduced the regular -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 on a ring , and they showed that is a general Krull ring if and only if each proper principal ideal of is written as a finite -product of prime ideals [8, Theorem 5.6].

Coincidentally, the -operation on a ring was also introduced by Tao [9], called the -operation in her master thesis independently. Unlike Chang and Oh who studied the -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 ring is Noetherian if and only if the direct sum of any number of injective -modules is injective, if and only if the direct sum of any countable number of injective -modules is injective, if and only if every injective -module is -injective, if and only if the direct limit of injective -modules is injective. An injective -module is said to be -injective if every direct sum of copies of is injective. In [10], Zhang et al. proved the -theoretic analog of the Cartan–Eilenberg–Bass theorem for Noetherian rings. The -Noetherian ring is defined to be the ring which satisfies the ascending chain condition (ACC) on -ideals by Yin et al. [11]. Also, a ring is -Noetherian if and only if satisfies the ACC on its -ideals (see [8, Theorem 6.9] or [9, Proposition 4.1.9]).

In [12], SM rings are introduced by Wang and Liao. A ring is called SM if satisfies the ACC on its regular -ideals [12, Definition 5.10]. SM rings can be regarded as regular -Noetherian ones loosely. Tao also proved that a ring is SM if and only if satisfies the ACC on its regular -ideals [9, Theorem 4.2.1]. Thus, a natural question arises as to whether SM rings have both -theoretic and -theoretic analogs corresponding to the Cartan–Eilenberg–Bass theorem for Noetherian rings. Actually for this purpose, the notions of reg-injective and -reg-injective -modules are introduced in [12]. An -module is reg-injective if for any regular ideal of [12, Definition 5.2]. A reg-injective -module 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 is SM if and only if the direct sum of any number of -torsion-free (or -torsion-free) reg-injective -modules is reg-injective, if and only if the direct sum of any countable number of -torsion-free (or -torsion-free) reg-injective -modules is reg-injective, if and only if every -torsion-free (or -torsion-free) reg-injective -module is -reg-injective. However, what about the direct limit of -torsion-free (or -torsion-free) reg-injective -modules for SM rings? Motivated by this question, we first show that the -operation can induce a torsion theory, denoted by , by a Gabriel topology

Also, for -Noetherian rings, we complete the -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 and -theoretic analogs of the Cartan–Eilenberg–Bass theorem for Noetherian rings. In the process, the discussion of the direct limit of -modules is also necessary.

2. Preliminaries

Now we introduce some notations and results needed in this paper from [9, 13]. Let be a finitely generated ideal of . If the natural homomorphism is an isomorphism, then is called a -ideal, denoted by . Let be an -module. Define

Thus, is a submodule of . Also, is said to be -torsion (resp., -torsion-free) if (resp., ). Clearly is a -torsion-free -module [11, Corollary 1.5]. A GV-torsion-free module is called a -module if for any . The -envelope of a -torsion-free module is the set given bywhere is the injective hull of . It is easy to see that is a -module if and only if . A nonzero ideal of is said to be a prime-ideal if is both a prime ideal and a -ideal and a maximal-ideal if is maximal in the set of all proper -ideals of . Note that each maximal -ideal is prime [13, Theorem 6.2.14]. A -torsion-free module is of finite type if for some finitely generated submodule of [13, Proposition 6.4.2]. A sequence of -modules and homomorphisms is said to be -exact if the sequence is exact for any maximal -ideal of . An -module is said to be of finitely presented type if there is a -exact sequence , where and are finitely generated and free modules [13, Definition 6.4.9].

An ideal of is regular if contains a nonzero divisor. An ideal of is called an-ideal if is a regular -ideal. Let denote the set of all -ideals of . Then, is a multiplicative system of ideals of , i.e., satisfies that (i) and (ii) if ; then, . It is clear that . But the converse does not hold in general.

Example 1. (see [9, Example 2.1.1]). Let be a field, and let and , where and are indeterminates over . Then, for the trivial extension , we have that . Then, . By [14, Theorem 4.7], is not a DW ring, i.e., the ring satisfies that every ideal of is a -ideal. Then, by [13, Theorem 6.3.12]. Thus, .
Let be an -module. DefineThus, is a submodule of . Also, is said to be rGV-torsion (resp., rGV-torsion-free) if (resp., ). It is clear that any -torsion-free -module is -torsion-free, while any -torsion -module is -torsion.

Example 2. (see [9, Example 2.2.4]). Let, but . Then, is-torsion, but not-torsion.

Proposition 1 (see [9, Proposition 2.2.6 and Proposition 2.2.7]). (1)An-moduleis-torsion if and only iffor any-torsion-free-module.(2)An-moduleis-torsion-free if and only iffor any-torsion-module.(3)Letbe a family of-modules. Then, is-torsion-free if and only if eachis-torsion-free.(4)Ifis an-torsion-free-module, thenis also-torsion-free.

An -torsion-free -module is called a-module if for any [9, Definition 3.1.1]. In [9, Definition 3.2.1], the -envelope of an -torsion-free -module is the set given by

It is clear that and that if and only if [9, Proposition 3.2.5].

Proposition 2 (see [9, Theorem 3.1.7 and Theorem 3.2.12]). The following statements are equivalent for an-torsion-free -module.(1)is a-module.(2).(3)Ifis an exact sequence in whichis a-module, thenis-torsion-free.(4)There exists an exact sequencesuch thatis a-module andis-torsion-free.(5)for any-torsion-module.(6)for any-torsion-module.

An ideal of is called a -ideal if is a -module. It is clear that -ideals of are -ideals, but the converse does not hold. One can refer to [8, Example 4.9]. A nonzero ideal of is said to be a prime -ideal if is both a prime ideal and a -ideal, denoted by -, and a maximal -ideal if is maximal in the set of all proper -ideals of , denoted by -. Note that each maximal -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)An-moduleis-torsion if and only iffor any maximal-ideal.(2)Letbe an-torsion-free-module. Then, .(3)Letbe an-torsion-free-module and letandbe submodules of. Then, if and only iffor any maximal-idealof.

An -module is said to be -finitely generated if there exists some finitely generated submodule of such that is an -torsion -module [9, Definition 3.4.1]. An -torsion-free -module is -finitely generated if and only if for some finitely generated submodule of [9, Proposition 3.4.3].

Proposition 4 (see [9, Proposition 3.4.5]). Letbe an exact sequence of-modules.(1)Ifandare-finitely generated, thenis-finitely generated.(2)Ifis-finitely generated, thenis-finitely generated.

3. Injective Modules over -Noetherian Rings

Next, as in [15], we show that the -operation can induce a torsion theory, denoted by , by a Gabriel topology

By the proof of [16, Proposition 4.6], the class of all -torsion -modules, denoted by , is the set { is an -module and for each nonzero element }. Let denote the set of all -torsion -modules. The following proposition shows that and coincide. Thus, -torsion-free (resp., -torsion) modules and -torsion-free (resp., -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 ring , .

Proof. Note that is an -torsion -module if and only if for any nonzero element there exists some such that ; if and only if for any nonzero element ; and if and only if is a -torsion -module.
Now we recall some terminology in [17], which is similar to that in [1515]. Let be an -module. A submodule of is called -pure (resp., -dense) in if is -torsion-free (resp., -torsion). Obviously if is a -dense submodule of an -torsion-free -module , then . Set , which is called the -closure of in . Then, is called -closed in if . It is easy to verify that if is -torsion-free, then ; is -dense in if and only if ; and -closed submodules of and its -pure submodules coincide.

Lemma 1. Let be an -torsion-free -module. If is a -module, then -pure submodules and -submodules of coincide.

Proof. Let be a submodule of . Then, the sequenceis exact for any . Note that if is a -pure submodule of , then is -torsion-free. So, . Obviously, because is a -module. Thus, = 0, and so is a -module. Conversely, if is a -module, then it is easy to prove that is -torsion-free. Thus, is a -pure submodule of .
Recall that an -module is said to be -Noetherian if satisfies the ACC on its -pure submodules [16, p. 175]. Thus, an -torsion-free -module is -Noetherian if satisfies the ACC on its -submodules by Lemma 1. Thus, an -torsion-free -Noetherian -module is also called -Noetherian in [9]. A ring is -Noetherian if is a -Noetherian -module. Note that is -torsion-free over . Then, is -Noetherian if and only if satisfies ACC on its -ideals.
In [9] or [8], it is proved that -Noetherian rings coincide with -Noetherian ones. Also, in terms of -operations, some characterizations of -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 ring.(1)is a-Noetherian ring.(2)is-Noetherian.(3)Every ideal is-finitely generated, i.e., for each idealof, there exists some finitely generated subidealofsuch that.(4)Every-ideal is-finitely generated.(5)Every nonempty set of-ideals ofhas a maximal element.(6)Every prime-ideal ofis-finitely generated over.(7)The direct sum of any number of-torsion-free injective-modules is injective.(8)The direct sum of any countable number of-torsion-free injective-modules is injective.(9)Every-torsion-free injective-module is-injective.

Remark. (1)In fact, with the help of the language of torsion theory, the equivalences of (2)–(6) of Proposition 6 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)Although -Noetherian rings coincide with -Noetherian ones, -Noetherian -modules are not necessarily -Noetherian. An -module is -Noetherian if satisfies ACC on its -submodules. If is a -submodule of , then is -torsion-free, which implies that is -torsion-free. Since , we can get that . Then, is a -pure submodule of . So, -Noetherian modules are -Noetherian by their definitions. But the converse does not hold by [9, Example 4.1.7]. In more detail, let and . Then, is -torsion, not -torsion over . Set . Thus, is an -torsion-free submodule of and is a -pure submodule of . Let be a direct sum of countably infinite number of . Then, is a -Noetherian -module since is -torsion. Note that the chain of -pure submodules of is not stationary. Then, is not a -Noetherian -module.Next, with the help of the language of torsion theory, we can get more characterizations of -Noetherian rings in terms of -operations. For this purpose, first we need some notions. An -module is -finitely generated if has a finitely generated -dense submodule [16, p. 157]. Thus, an -torsion-free -module is -finitely generated (also -finitely generated in this case) if there exists a finitely generated submodule of such that , equivalently . An -module is -finitely presented if it is isomorphic to , where is a finitely generated free -module and is a -finitely generated submodule of [16, p. 164]. A ring is -coherent if every finitely generated ideal of is -finitely presented [18, Definition 1.2]. From [18, Theorem 3.3], we can get that is a -coherent ring if and only if the direct limit of -torsion-free FP-injective -modules is FP-injective. Recall that an -module is said to be FP-injective if for all finitely presented -modules . It can be also called an absolutely pure-module. For more details, one can refer to [16].
Now, based on Proposition 6, we complete the -version of the Cartan–Eilenberg–Bass theorem for Noetherian rings. For this, we need the following.

Lemma 2. Letbe a family of rGV-torsion-free-modules. Then, andare also-torsion-free.

Proof. Let be an exact sequence, where . Then, is finitely presented. Thus, for any by [10, Lemma 2.1]. So, is -torsion-free. Since is a submodule of , which is -torsion-free by Proposition 1 (3), it is clear that is -torsion-free.

Theorem 1. The following statements are equivalent for a ring.(1)is a-Noetherian ring.(2)Every finitely generated-module is-finitely presented.(3)Every finitely generated-module is-Noetherian.(4)The direct limit of-torsion-free injective-modules is injective.(5)-torsion-free FP-injective-modules and-torsion-free injective ones coincide.

Proof. Let be a finitely generated -module. Then, there exists an exact sequence , where is a finitely generated free -module. Since -Noetherian rings and -Noetherian ones coincide, it follows that is a -Noetherian -module by [16, Proposition 20.4]. Then, is -finitely generated [16, Proposition 20.1]. Thus, is -finitely presented by [16, Proposition 19.3]. Let be an ideal of . Since is finitely generated over , it is -finitely presented by (2). Then, is -finitely generated by [16, Proposition 19.3]. Thus, is a -Noetherian ring again by [16, Proposition 20.1] or [17, Proposition 2.3.3]. Then, is -Noetherian. Let be a finitely generated -module. Then, there exists an exact sequence , where is a finitely generated free -module. By [16, Proposition 20.4], is a -Noetherian -module. Again by [16, Proposition 20.4], is also a -Noetherian -module. It is clear. The proof of this implication is very similar to that of in [15, Theorem 3.17]; however, we give a proof for completeness. Assume that is a -Noetherian ring. For any ideal of , is -finitely generated over by Proposition 6. Thus, there exists a finitely generated subideal of such that . So, is an -torsion -module. Let be an -torsion-free FP-injective -module. Then, for the exact sequence , we can get the following exactsequence: Since is -torsion over and is -torsion-free over , it follows that by Proposition 1 (1). Also, since is FP-injective, we can get that . Then, . Thus, is an injective -module. Note that the direct sum of FP-injective -modules is FP-injective by [19, p. 564]. Then, (1) holds by the equivalence of (1) and (7) in Proposition 6. By (2), is -coherent. Then, we can get that the direct limit of -torsion-free FP-injective -modules is FP-injective by [4, Theorem 3.3]. By (6) and Lemma 2, the direct limit of -torsion-free injective -modules is injective. Since -torsion-free -modules are -torsion-free, we can get that the direct limit of -torsion-free injective modules is injective by (4). Then, is -Noetherian by [10, Theorem 2.9].

4. The Direct and Inverse Limits of -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 of -modules is necessary [20]. First we show that any -ideal of is -finitely presented.

Let be an -module, , and . Then, we have the natural homomorphism by

Lemma 3 (see [13, Theorem 2.6.17]). Letbe an-module. Then, is finitely generated projective if and only ifis an isomorphism.

Lemma 4. Letbe an-torsion-free-module and letbe as in the above. Ifis an isomorphism for any-, thenis-finitely generated.

Proof. Since is -torsion-free, it is easy to see that is also -torsion-free. Note that by considering that is an isomorphism for any -. Then, by Proposition 3. Thus, there exists some such that , where denotes the identity map on . Set . Then, for any , there are finite sets and such that . Let be the submodule of generated by . Then, for any , we have . Thus, , which implies that . It is clear that . Then, . Therefore, is -finitely generated.

Lemma 5. Let. Then, is-torsion for any-module.

Proof. Let . For the exact sequence , we can get an exact sequencewhere is an -module. Note that is an -module and so an -torsion -module. Then, is -torsion over .
Let be a multiplicatively closed set of . An -module is said to be -torsion if , and that is said to be -torsion-free if , for and , implies .

Lemma 6. Letand-. Then, for any -module.

Proof. Set and set . Then, is -torsion and is -torsion-free. For the exact sequence , we can get the following exact sequence:By Lemma 1, . Note that is -torsion. Then, . Thus, . By [21, Lemma 1.7], . Note that . Then, .

Theorem 2. Any -ideal of is -finitely presented.

Proof. Let . Then, there exits an exact sequence , where is finitely generated free module. Let be a maximal -ideal of . Then, for any -module by Lemma 1. Note that . Then, . Thus, we have the following commutative diagram (12) with exact rows:Note that is an isomorphism by Lemma 5 and is an isomorphism by [13, Theorem 2.6.16 (1)]. Then, . In particular, and . For the exact sequence , since and are finitely generated and free over , we can get that is also finitely generated free over .
Consider the following commutative diagram:By Lemma 3, 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, is a -isomorphism. Note that is an -torsion-free -module. Then, is -finitely generated over by Lemma 4. Thus, is -finitely presented over .
Next, we show that the direct and inverse limits of -modules are also -modules. To do so, we need the following.

Lemma 7. Let be a family of -torsion-free -modules. Then, for each.

Proof. Let . Since is finitely generated, there is a finitely generated free module such that is an exact sequence of -modules. Note that is -finitely presented by Theorem 2-, and so is -finitely generated. Since is -torsion-free, we can get that there exists a finitely generated submodule of such that . Then, there is an exact sequence , where is a finitely generated free -module. Thus, we can get an exact sequence . For any maximal -ideal , . So, is -torsion. Then, we have the following commutative diagram (14) with exact rows:By Lemma 2, is -torsion-free. Then, it follows from Proposition 1 (1) thatBy [10, Lemma 2.1], is an isomorphism. Then, is a monomorphism by the Five Lemma.
Consider the following commutative diagram (16):By the above proof, is a monomorphism. Again by [10, Lemma 2.1], is an isomorphism. Hence, is an epimorphism by the Five Lemma and is a monomorphism also by the above proof. Therefore, is an isomorphism.

Corollary 1. Let be a family of -torsion-free -modules. If is -torsion-free and -finitely generated, thenis a monomorphism.

Theorem 3. Letbe a family of -modules over. Then, is a-module.

Proof. Let . Then, is an exact sequence. Consider the following commutative diagram:By Lemma 7, is an isomorphism. Then, is an isomorphism by the Five Lemma. Since each is a -module, . Hence, is a -module.

Lemma 8 (see [22, Theorem 2.22]). The inverse limit of an inverse systemof modules exists and.

Theorem 4. Letbe an inverse system of-modules over. Then, is a -module.

Proof. It suffices to show that every -homomorphism can be extended to for each .
Let be an embedding and let be an embedding. Consider the following diagram:Since is a -module, there is a homomorphism such that . Set . For each , . Then, whenever by Lemma 8. Hence, . Because is -torsion-free, we can get that , which implies whenever . Thus, again by Lemma 8. Therefore, is a -module.

5. Reg-Injective Modules over SM Rings

In this section, for SM rings, we complete both the and the -theoretic analogs of the Cartan–Eilenberg–Bass theorem for Noetherian rings. In [9], some characterizations are given in terms of -operations.

Proposition 7 (see [9, Theorem 4.2.1]). The following statements are equivalent for a ring.(1)is an SM ring.(2)satisfies ACC on its regular-ideals.(3)Every regular-ideal ofis-finitely generated.(4)Every nonempty set of regular-ideals ofhas a maximal element.(5)Every regular prime-ideal ofis-finitely generated.(6)Every regular ideal ofis-finitely generated.

Theorem 5. Letbe an SM ring. Then, every finitely generated regular ideal ofis-finitely presented.

Proof. Let be a finitely generated regular ideal of , where is a regular element in . We prove this by induction on . If , then , which implies that is -finitely presented. Now assume that is -finitely presented, where . Then, we have the following commutative diagram (20) with exact rows:where for any and
Now we explain why. Define by for any . Then, it is easy to verify that is well defined. For any , . Then, there exists some elements such that . Thus, and , which implies that is an epimorphism. It is clear that . For any , . Then, . Then, . Thus, . So,.
Note that . Then, is a regular ideal of . Since is an SM ring, we can get that is -finitely generated by Proposition 7. By assumption, is -finitely presented, and so is -finitely generated [3, Proposition 19.3]. Thus, is -finitely generated by Proposition 4. Hence, is -finitely presented.
It is well known that Noetherian rings are coherent ones, and -Noetherian rings are -coherent ones [13, p. 393]. Recall that is -coherent if each finite type ideal of is of finitely presented type [13, Definition 6.9.14]. By the same way of Theorem 5, we can get that if is an SM ring, then every finitely generated regular ideal of is of finitely presented type, where such a ring is called regular-coherent [7, Definition 2.4]. Then, we can get the following corollary, which is corresponding to the classical result.

Corollary 2. SM rings are regular-coherent.

Next, our purpose is to complete the Cartan–Eilenberg–Bass theorem for SM rings. To do so, we need the following.

Lemma 9. Letbe a family of-torsion-free-modules. Ifis an SM ring, thenfor any finitely generated regular idealof.

Proof. Let be a finitely generated regular ideal of . Since is an SM ring, we can get that is -finitely generated by Theorem 5. Then, there exists an exact sequence , where is finitely generated and free over and is -finitely generated. Thus, we have the following commutative diagram (21) with exact rows:By Corollary 1, is a monomorphism. By [10, Lemma 2.1], is an isomorphism. Then, is an epimorphism by the Five Lemma. Note that is a monomorphism again by Corollary 1. Thus, is an isomorphism.

Lemma 10. Letbe an-torsion-free-module and letbe a-module. Then, .

Proof. For the exact sequence , we can get an exact sequence . Note that is -torsion and is a -module. Then, by Proposition 1 (1) and Proposition 2. Thus, .

Proposition 8. Letbe a family of-modules. Ifis an SM ring, thenfor any regular-idealof.

Proof. Let be a regular -ideal of . From the exact sequence , we have the following commutative diagram (22) with exact rows:By [10, Lemma 2.1], is an isomorphism. Since is SM, we can get that is -finitely generated by Proposition 7. Then, there exists a finitely generated subideal of such that . It is easy to verify that is regular. Thus, by Lemma 10. by Lemma 9. by Theorem 3 and Lemma 10. Then, .
Recall that an -module is reg-injective if for any regular ideal of [12, Definition 5.2].

Proposition 9. Letbe a commutative ring and letbe a-module over. Then, the following statements are equivalent.

(1) is a reg-injective -module.(2) for any regular -ideal of .(3)For any regular -ideal of , every homomorphism can be extended to .

Proof. This is trivial. Let be a regular ideal of . Then, is a regular ideal of and is -torsion. Since is a -module, we can get that by Proposition 2. From the exact sequence , we can get an exact sequence Thus, . Therefore, is injective. This is clear. Recall that a reg-injective -module 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 ring.

(1) is an SM ring.(2)The direct sum of any number of -torsion-free reg-injective -modules is reg-injective.(3)The direct sum of any number of -torsion-free reg-injective -modules is reg-injective.(4)The direct sum of any countable number of -torsion-free reg-injective -modules is reg-injective.(5)The direct sum of any countable number of -torsion-free reg-injective -modules is reg-injective.(6)Every -torsion-free reg-injective -module is -reg-injective.(7)Every -torsion-free reg-injective -module is -reg-injective.(8)The direct limit of -torsion-free reg-injective -modules is reg-injective.(9)The direct limit of -torsion-free reg-injective -modules is reg-injective.

Proof. See [9, Theorem 4.1.10]. See [12, Theorem 6.10]. Let be a family of -torsion-free reg-injective -modules. Then, each is a -module by definition. Since is an SM ring, we can get that for any regular -ideal of by Proposition 8. Note that is a -module by Theorem 3. Then, is reg-injective by Proposition 9. This follows by the fact that -torsion-free -modules are -torsion-free. This follows by the fact that a direct sum is a direct limit of finite sums.

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Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

Hwankoo Kim was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2021R1I1A3047469). De Chuan Zhou was supported by the National Natural Science Foundation of China (no. 12101515). Xiao Lei Zhang was supported by the National Natural Science Foundation of China (no. 12061001).

References

R. Kennedy, “Krull rings,” Pacific Journal of Mathematics, vol. 89, no. 1, pp. 131–136, 1980.
View at: Publisher Site | Google Scholar
R. Matsuda, “Notes of prüfer v-multiplication rings,” Bulletin of the Faculty of Science, vol. 12, pp. 9–15, 1980.
View at: Google Scholar
B. G. Kang, “A characterization of Krull rings with zero divisors,” Journal of Pure and Applied Algebra, vol. 72, no. 1, pp. 33–38, 1991.
View at: Publisher Site | Google Scholar
B. G. Kang, “Characterizations of Krull rings with zero divisors,” Journal of Pure and Applied Algebra, vol. 146, no. 3, pp. 283–290, 2000.
View at: Publisher Site | Google Scholar
T. G. Lucas, “Krull rings, Prüfer v-multiplication rings and the ring of finite fractions,” Rocky Mountain Journal of Mathematics, vol. 35, no. 4, pp. 1251–1326, 2005.
View at: Google Scholar
F. G. Wang and H. Kim, “w-injective modules and w-semi-hereditary rings,” Journal of the Korean Mathematical Society, vol. 51, no. 3, pp. 509–525, 2014.
View at: Google Scholar
X. L. Zhang, “A homological characterization of Prüfer v-multiplication rings,” Bull. Korean Math. Soc., , vol. 59, no. 1, pp. 213–226, 2022.
View at: Google Scholar
G. W. Chang and J. S. Oh, “Prime factorization of ideals in commutative rings, with a focus on Krull rings,” Journal of Korean Mathematical Society, 2022, https://arxiv.org/abs/2204.12098.
View at: Google Scholar
Y. Z. Tao, Research On the w+-Module Over Commutative Rings, Sichuan Normal University, Sichuan, China, 2022, Master Thesis.
J. Zhang, F. G. Wang, and H. Kim, “Injective modules over w-Noetherian rings, II,” Journal of the Korean Mathematical Society, vol. 50, no. 5, pp. 1051–1066, 2013.
View at: Publisher Site | Google Scholar
H. Y. Yin, F. G. Wang, X. S. Zhu, and Y. H. Chen, “w-Modules over commutative rings,” Journal of the Korean Mathematical Society, vol. 48, no. 1, pp. 207–222, 2011.
View at: Publisher Site | Google Scholar
F. G. Wang and J. L. Liao, “S-injective modules and S-injective envelopes,” Acta Mathematica Sinica, vol. 54, no. 2, pp. 271–284, 2011.
View at: Google Scholar
F. G. Wang and H. Kim, Foundations of Commutative Rings and Their Modules, Springer, Berlin, Germany, 2016.
F. G. Wang, D. C. Zhou, and D. Chen, “Module-theoretic characterizations of the ring of finite fractions of a commutative ring,” Journal of Commutative Algebra, vol. 14, no. 1, pp. 141–154, 2022.
View at: Publisher Site | Google Scholar
D. C. Zhou, H. Kim, F. G. Wang, and D. Chen, “A new semistar operation on a commutative ring and its applications,” Communications in Algebra, vol. 48, no. 9, pp. 3973–3988, 2020.
View at: Publisher Site | Google Scholar
J. S. Golan, “Torsion Theories,” Pitman Monographs and Surveys in Pure and Applied Mathematics, Longman Scientific and Technical, Horlow, 1986.
View at: Google Scholar
P. E. Bland, Topics in Torsion Theory, Wiley-VCH, Weinheim, Germany, 1998.
M. F. Jones, “Coherent relative to an hereditary torsion theory,” Communications in Algebra, vol. 10, no. 7, pp. 719–739, 1982.
View at: Publisher Site | Google Scholar
C. Megibben, “Absolutely pure modules,” Proceedings of the American Mathematical Society, vol. 26, no. 4, pp. 561–566, 1970.
View at: Publisher Site | Google Scholar
D. C. Zhou and F. G. Wang, “The direct and inverse limits of w-modules,” Communications in Algebra, vol. 44, no. 6, pp. 2495–2500, 2016.
View at: Google Scholar
F. G. Wang and H. Kim, “Two generalizations of projective modules and their applications,” Journal of pure and applied algebra, vol. 219, no. 6, pp. 2099–2123, 2015.
View at: Publisher Site | Google Scholar
J. J. Rotman, An Introduction to Homological Algebra, Springer, Berlin, Germany, 1979.

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Copyright © 2022 De Chuan Zhou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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