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Kyungpook Mathematical Journal 2023; 63(1): 29-36

Published online March 31, 2023 https://doi.org/10.5666/KMJ.2022.63.1.29

Copyright © Kyungpook Mathematical Journal.

On Two Versions of Cohen's Theorem for Modules

Xiaolei Zhang and Wei Qi, Hwankoo Kim∗

School of Mathematics and Statistics, Shandong University of Technology, Zibo 255049, China
e-mail : zxlrghj@163.com and qwrghj@126.com

Division of Computer Engineering, Hoseo University, Asan 31499, Republic of Korea
e-mail : hkkim@hoseo.edu

Received: October 16, 2021; Revised: July 25, 2022; Accepted: August 8, 2022

Parkash and Kour obtained a new version of Cohen's theorem for Noetherian modules, which states that a finitely generated R-module M is Noetherian if and only if for every prime ideal 𝔭 of R with Ann(M) 𝔭, there exists a finitely generated submodule N𝔭 of M such that 𝔭MN𝔭M(𝔭), where M(𝔭)={xMsx 𝔭M for some sR 𝔭}. In this paper, we generalize the Parkash and Kour version of Cohen's theorem for Noetherian modules to S-Noetherian modules and w-Noetherian modules.

Keywords: Cohen's theorem, S-Noetherian modules, w-Noetherian modules.

Throughout this article, all rings are commutative rings with identity and all modules are unitary. Let R be a ring and M an R-module. For a subset U of M, we denote by U the submodule of M generated by U. Early in 1950, Cohen showed that a ring R is Noetherian if and only if every prime ideal of R is finitely generated [4][Theorem 2]. Let 𝔭 be a prime ideal of R. Following [10], we set M(𝔭):={xMsx𝔭M for some sR𝔭}. Then M(𝔭) is obviously a submodule of M. In 1994, Smith extended Cohen's theorem from rings to modules, showing that a finitely generated R-module M is Noetherian if and only if the submodules 𝔭M of M are finitely generated for every prime ideal 𝔭 of R, if and only if M(𝔭) is finitely generated for each prime ideal 𝔭 of R with 𝔭 Ann(M) [12]. Recently, Parkash and Kour generalized Smith's result on Noetherian modules as follows:

Theorem. ([11, Theorem 2.1.]) Let R be a ring and M a finitely generated R-module. Then M is Noetherian if and only if for every prime ideal 𝔭 of R with Ann(M)𝔭, there exists a finitely generated submodule N𝔭 of M such that 𝔭MN𝔭M(𝔭).

In the past few decades, some generalizations of Noetherian rings or Noetherian modules have been extensively studied, especially via some multiplicative subsets S of R and the w-operation (see [1, 6, 7, 8, 9, 15, 16] for example). And the related Cohen's theorem has also been considered by many authors (see [1, 3, 5, 14] for example). In 2002, Anderson and Dumitrescu gave an analogue of Cohen's theorem for S-Noetherian modules, which states that an S-finite module M is S-Noetherian if and only if the submodules of the form 𝔭M are S-finite for each prime ideal 𝔭 of R (disjoint from S) [1, Proposition 4]. In 1997, Wang and McCasland obtained an analogue of Cohen's theorem for strong Mori (SM) modules M over integer domains for which M satisfies the ascending chain condition on w-submodules of M. In fact, they showed that a w-module M is an SM module if and only if each w-submodule of M is w-finite type, if and only if M and every prime w-submodule of M are w-finite type [14, Theorem 4.4]. In this paper, we give both an S-analogue and a w-analogue of Parkash and Kour's result on Noetherian modules, which can be seen as generalizations of Cohen's theorem for modules.

Let R be a ring and S a multiplicative subset of R, that is 1 ∈ S and s1s2S for any s1∈ S, s2S. Let M be an R-module. Recall from [1] that M is called S-finite if sMF for some s∈ S and some finitely generated submodule F of M. Also, M is called S-Noetherian if each submodule of M is an S-finite R-module. Then R is called an S-Noetherian ring if R is S-Noetherian as an R-module. Anderson and Dumitrescu obtained a Cohen-type theorem for S-Noetherian modules: An S-finite R-module M is S-Noetherian if and only if the submodules of the form 𝔭M are S-finite for each prime ideal 𝔭 of R (disjoint from S) [1][Proposition 4]. Now we give a "stronger" version of Cohen's theorem for S-Noetherian modules which can be seen as an S-analogue of Parkash and Kour's result [11, Theorem 2.1].

Theorem 2.1. Let R be a ring and S a multiplicative subset of R. Let M be an S-finite R-module. Then M is S-Noetherian if and only if for every prime ideal 𝔭 of R with Ann(M)𝔭, there exists an S-finite submodule N𝔭 of M such that 𝔭MN𝔭M(𝔭).

Proof. Suppose that M is an S-Noetherian R-module and let 𝔭 be a prime ideal with Ann(M)𝔭. If we take N𝔭:=𝔭M, then N𝔭 is certainly an S-finite submodule of M satisfying 𝔭MN𝔭M(𝔭).

Conversely, suppose on the contrary that M is not S-Noetherian. Let N be the set of all submodules of M which are not S-finite. Then N is non-empty. Make a partial order on N by defining N1N2 if and only if N1N2 in N. Let {NiiΛ} be a chain in N. Set N:= iΛNi. Then N is not S-finite. Indeed, suppose sNx1,,xnN for some s∈ S. Then there exists i0Λ such that {x1,,xn}Ni0. Thus sNi0sNx1,,xnNi0 implies that Ni0 is S-finite, which is a contradiction.

Then by Zorn's Lemma, N has a maximal element, which is also denoted by N. Set 𝔭:=(N:M)={rRrMN}.

We claim that 𝔭 is a prime ideal of R. Assume on the contrary that there exist a,bR𝔭 such that ab𝔭. Since a,bR𝔭, we have aMN and bMN. Therefore N+aM is S-finite. Let {y1,,ym} be a subset of N+aM such that s1(N+aM)y1,,ym for some s1S. Write yi=wi+azi for some wiN and ziM(1im). Set L:={xMaxN}. Then N+bML, and hence L is also S-finite. Let {x1,,xk} be a subset of L such that s2Lx1,,xk for some s2S. Let n ∈ N and write

s1n= i=1mriyi= i=1mriwi+a i=1mrizi.

Then i=1mriziL. Thus s2 i=1mrizi= i=1k rixi for some riR (i=1,,k).

So s1s2n= i=1ms2riwi+ i=1k riaxi. Thus s1s2Nw1,,wm,ax1,,axk implies that N is S-finite, which is a contradiction.

We also claim that M(𝔭)N. Suppose on the contrary that there exists yM(𝔭) such that yN. Then there exists tR𝔭 such that ty𝔭M=(N:M)MN. As t𝔭=(N:M), it follows that tMN. Therefore N+tM is S-finite. Let {u1,,um} be a subset of N+tM such that s3(N+tM)u1,,um for some s3S. Write ui=wi+tzi (i=1,,m) with wiN and ziM. Set T:={xMtxN}. Then NN+RyT, and hence T is S-finite. Then there exists a subset {v1,,vl} of T such that s4Tv1,,vl for some s4 ∈ S. Let n be an element in N. Then

s3n= i=1mriui= i=1mriwi+t i=1mrizi.

Thus i=1mriziT. So s4 i=1mrizi= i=1l rivi for some riR (i=1,,l). Hence s3s4n= i=1ms4riwi+ i=1l ritvi. Thus s3s4Nw1,,wm,tv1,,tvl implies that N is S-finite, which is a contradiction.

Let F=m1,,mk be a submodule of M such that sMF for some s∈ S. Claim that 𝔭S=. Indeed, if s𝔭 for some s'∈ S, then sMNM. So ssNssMsFsMN implies that N is S-finite, which is a contradiction. Note that 𝔭=(N:M)(N:F)(N:sM)=(𝔭:s)=𝔭 as 𝔭 is a prime ideal of R. So 𝔭=(N:F)=(N:m1,,mk)= i=1 k(N:Rmi). By [2][Proposition 1.11], 𝔭=(N:Rmj) for some 1jk. Since mjN, it follows that N+Rmj is S-finite. Let {y1,,ym} be a subset of N+Rmj such that s5(N+Rmj)y1,,ym for some s5S. Write yi=wi+aimj for some wiN and aiR (i=1,,m). Let n ∈ N. Then s5n= i=1mri(wi+aimj)= i=1mriwi+( i=1mriai)mj. Thus ( i=1mriai)mjN. So i=1mriai𝔭. Thus s5Nw1,,wm+𝔭mj. As Ann(M)(N:M)=𝔭, there exists an S-finite submodule N𝔭 of M such that 𝔭MN𝔭M(𝔭). Thus

s5Nw1,,wm+𝔭mj  w1,,wm+𝔭M  w1,,wm+N𝔭  w1,,wm+M(𝔭)  N

Since N𝔭+w1,,wm is S-finite, it follows that N is also S-finite, which is a contradiction. Hence M is S-Noetherian.

Taking S={1}, we can recover the following result of Parkash and Kour.

Corollary 2.2. ([11, Theorem 2.1]) Let R be a ring and M a finitely generated R-module. Then M is Noetherian if and only if for every prime ideal 𝔭 of R with Ann(M)𝔭, there exists a finitely generated submodule N𝔭 of M such that 𝔭MN𝔭M(𝔭).

We recall some basic knowledge on the w-operation over a commutative ring. One can refer to [13] for more details. Let R be a commutative ring and J a finitely generated ideal of R. Then J is called a GV-ideal if the natural homomorphism R HomR(J,R) is an isomorphism. The set of GV-ideals is denoted by GV(R). Let M be an R-module. Define

tor(M):={xMJx=0 for some JGV(R)}.

An R-module M is said to be GV-torsion (resp., GV-torsion-free) if tor(M)=M (resp., tor(M)=0). A GV-torsion-free module M is called a {w-module} if ExtR1(R/J,M)=0 for any J GV(R). A DW ring R is a ring for which every R-module is a w-module.

A maximal w-ideal is an ideal of R which is maximal among the w-submodules of R. The set of all maximal w-ideals is denoted by w-Max(R). Each maximal w-ideals is a prime ideal (see {[13, Theorem 6.2.14]).

An R-homomorphism f:MN is said to be a w-monomorphism (resp., w-epimorphism, w-isomorphism) if for any 𝔭wMax(R), f𝔭:M𝔭N𝔭 is a monomorphism (resp., an epimorphism, an isomorphism). Note that f is a w-monomorphism (resp., w-epimorphism) if and only if Ker(f) (resp., Coker(f)) is GV-torsion. An R-module M is said to be w-finite type if there exist a finitely generated free module F and a w-epimorphism g:FM. Obviously, an R-module M is w-finite type if and only if there is a finitely generated submodule N of M such that M/N is GV-torsion.

Lemma 3.1. Let N be a w-submodule of a GV-torsion-free w-finite type module M. Then (N:RM)𝔭=(N𝔭:R𝔭M𝔭) for any prime w-ideal 𝔭 of R.

Proof. Let 𝔭 be a prime w-ideal of R. Obviously, (N:RM)𝔭(N𝔭:R𝔭M𝔭). On the other hand, since M is a w-finite type R-module, there exists a finitely generated submodule F=m1,,mn of M satisfying that for any m∈ M there exists J∈ΓV(R) such that Jm⊆ F. Let rs be an element in (N𝔭:R𝔭M𝔭). Then for each i=1,,n, there exists siR𝔭 such that sirmiN. Thus s1snrFN. So s1snrJmN for all mME(M), where E(M) is the injective envelope of M. By [13, Theorem 6.16], s1snrMN since N is a w-module. Hence s1snr(N:RM). Consequently, rs=s1snrs1sns(N:RM)𝔭.

Let M be an R-module. Recall from [13, Definition 8.1] that M is called a w-Noetherian module if every submodule of M is w-finite type. And R is called a w-Noetherian ring if R is w-Noetherian as an R-module.

Theorem 3.2. Let R be a ring and M a GV-torsion-free w-finite type R-module. Then M is a w-Noetherian module if and only if for every prime w-ideal 𝔭 of R with Ann(M)𝔭, there exists a w-finite type submodule N𝔭 of M such that 𝔭MN𝔭M(𝔭).

Proof. Suppose that M is a w-Noetherian R-module and let 𝔭 be a prime w-ideal with Ann(M)𝔭. If we take N𝔭:=𝔭M, then N𝔭 is certainly a w-finite type submodule of M satisfying 𝔭MN𝔭M(𝔭).

Conversely, suppose on the contrary that M is not w-Noetherian. Let N be the set of all w-submodules of M which are not w-finite type. Then N is non-empty. Make a partial order on N by defining N1N2 if and only if N1N2 in N. Let {NiiΛ} be a chain in N. Set N:= iΛNi. Then N is not w-finite type. Indeed, suppose there is an exact 0FNT0 with T GV-torsion and F=x1,,xn finitely generated. Then there exists i0Λ such that FNi0. Consider the following commutative diagram with exact rows:

Since T' is a submodule of T, we have that T' being GV-torsion implies that Ni0 is w-finite type, which is a contradiction. Since N is a w-submodule of M, it follows that NN. So by Zorn's Lemma, N has a maximal element, which is also denoted by N. Set 𝔭:=(N:M)={rRrMN}. Then 𝔭 is a w-ideal by [13, Section 6.10, Exercise 6.8].

We claim that 𝔭 is a prime ideal of R. Assume on the contrary that there exist a,bR𝔭 such that ab𝔭. Since a,bR𝔭, we have aMN and bMN. Therefore N+aM is w-finite type. Let {y1,,ym} be a subset of N+aM such that 0F1N+aMT10 be an exact sequence with T1 GV-torsion and F1=y1,,ym finitely generated. Write yi=wi+azi for some wiN and ziM(1im). Set L:={xMaxN}. Then N+bML, and hence L is w-finite type. Let 0F2LT20 be an exact sequence with T2 GV-torsion and F2=x1,,xk finitely generated. Let n be an element in N. Then there is a GV-ideal J1=j11,,j1p such that J1nF1. So there is {ritt=1,,p;i=1,,m}R such that

j1tn= i=1mrityi= i=1mritwi+a i=1mritzi(t=1,,p).

Then i=1mritziL (t=1,,p). Thus there exists a GV-ideal J2=j21,,j2l such that j2s i=1mritzi= i=1k rit,sxi for some {rit,si=1,,k;t=1,,p;s=1,,l}R.

So j1tj2sn= i=1mj2sritwi+ i=1k rit,saxi(t=1,,k;s=1,,l). Thus

J1J2nw1,,wm,ax1,,axk

implies that N is w-finite type, which is a contradiction.

We claim that M(𝔭)N. Assume on the contrary that there exists an element yM(𝔭) such that yN. Then there exists tR𝔭 such that ty𝔭M=(N:M)MN. As t𝔭=(N:M), it follows that tMN. Therefore N+t'M is w-finite type. Let 0F3N+tMT30 be an exact sequence with T3 GV-torsion and F3=u1,,um a finitely generated submodule of N+t'M. Write ui=wi+tzi (i=1,,m) with wi∈ N and zi∈ M. Set L:={xMtxN}. Then NN+RyL, and hence L is w-finite type. Let 0F4LT40 be an exact sequence with T4 GV-torsion and F4=u1,,un a finitely generated submodule of L. Let n be an element in N. Then there is a GV-ideal J3=j31,,j3k such that J3nF3. So there is {ritt=1,,p;i=1,,m}R such that

j3tn= i=1mritui= i=1mritwi+t i=1mritzi(t=1,,p).

So i=1mritziL(t=1,,p). Thus there exists a GV-ideal J4=j41,,j4l such that j4s i=1mritzi= i=1n rit,sui for some {rit,si=1,,m;t=1,,p;s=1,,l}R. So j3tj4sn= i=1mj4sritwi+ i=1k rit,stui(t=1,,k;s=1,,l). Thus J3J4nw1,,wm,tu1,,tuk implies that N is w-finite type, which is a contradiction.

Let 𝔪 be a maximal w-ideal of R and F=m1,,mk a submodule of M such that M/F is GV-torsion. So M𝔪=F𝔪. Then (N:RM)𝔪=(N𝔪:R𝔪M𝔪)=(N𝔪:R𝔪F𝔪)=(N:RF)𝔪 by Lemma 3.1. By [13, Section 6.10, Exercise 6.8], (N:RM) and (N:RF) are all w-ideals. So we have 𝔭=(N:RM)=(N:RF)= i=1k(N:Rmi). By [2][Proposition 1.11], 𝔭=(N:RRmj) for some 1jk. Since mjN, it follows that N+Rmj is w-finite type. Let 0F5N+RmjT50 be an exact sequence with T5 GV-torsion and F5=y1,,ym a finitely generated submodule of N+Rmj. Write yi=wi+aimj for some wi∈ N and aiR (i=1,,m). Let n be an element in N. Then there is a GV-ideal J5=j51,,j5l such that J5nF5. So there is {ritt=1,,p;i=1,,m}R such that j5tn= i=1mrityi= i=1mritwi+( i=1mritai)mj(t=1,,l). So i=1mritai𝔭. Thus J5Nw1,,wm+𝔭mj. As Ann(M)(N:M)=𝔭, there exists a w-finite type submodule N𝔭 of M such that 𝔭MN𝔭M(𝔭). Thus

J5Nw1,,wm+𝔭mj  w1,,wm+𝔭M  w1,,wm+N𝔭  w1,,wm+M(𝔭)  N

Since N𝔭+w1,,wm is w-finite type, it follows that N is also w-finite type, which is a contradiction. Hence M is w-Noetherian.

Taking M:=R, we have the following characterization of w-Noetherian rings.

Corollary 3.3. ([15, Theorem 4.7(1)]) Let R be a ring. Then R is a w-Noetherian ring if and only if each prime w-ideal of R is w-finite type.

The first author was supported by the National Natural Science Foundation of China (No. 12061001). The second author was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education (2021R1I1A3047469).

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