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):={x∈M∣sx∈𝔭M for some s∈R∖𝔭}. 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 𝔭M⊆N𝔭⊆M(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 s1s2∈S for any s₁∈ S, s2∈S. Let M be an R-module. Recall from [1] that M is called S-finite if sM⊆F 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 𝔭M⊆N𝔭⊆M(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 𝔭M⊆N𝔭⊆M(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 N1≤N2 if and only if N1⊆N2 in N. Let {Ni∣i∈Λ} be a chain in N. Set N:=∪ i∈ΛNi. Then N is not S-finite. Indeed, suppose sN⊆⟨x1,…,xn⟩⊆N for some s∈ S. Then there exists i0∈Λ such that {x1,…,xn}⊆Ni0. Thus sNi0⊆sN⊆⟨x1,…,xn⟩⊆Ni0 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)={r∈R∣rM⊆N}.

We claim that 𝔭 is a prime ideal of R. Assume on the contrary that there exist a,b∈R∖𝔭 such that ab∈𝔭. Since a,b∈R∖𝔭, we have aM⊆N and bM⊆N. Therefore N+aM is S-finite. Let {y1,…,ym} be a subset of N+aM such that s1(N+aM)⊆⟨y1,…,ym⟩ for some s1∈S. Write yi=wi+azi for some wi∈N and zi∈M (1≤i≤m). Set L:={x∈M∣ax∈N}. Then N+bM⊆L, and hence L is also S-finite. Let {x1,…,xk} be a subset of L such that s2L⊆⟨x1,…,xk⟩ for some s2∈S. Let n ∈ N and write

Then ∑ i=1mrizi∈L. Thus s2∑ i=1mrizi=∑ i=1k r′ixi for some r′i∈R (i=1,…,k).

So s1s2n=∑ i=1ms2riwi+∑ i=1k r′iaxi. Thus s1s2N⊆⟨w1,…,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 y∈M(𝔭) such that y∈N. Then there exists t∈R∖𝔭 such that ty∈𝔭M=(N:M)M⊆N. As t∈𝔭=(N:M), it follows that tM⊆N. Therefore N+tM is S-finite. Let {u1,…,um} be a subset of N+tM such that s3(N+tM)⊆⟨u1,…,um⟩ for some s3∈S. Write ui=wi+tzi (i=1,…,m) with wi∈N and zi∈M. Set T:={x∈M∣tx∈N}. Then N⊂N+Ry⊆T, and hence T is S-finite. Then there exists a subset {v1,…,vl} of T such that s4T⊆⟨v1,…,vl⟩ for some s₄ ∈ S. Let n be an element in N. Then

Thus ∑ i=1mrizi∈T. So s4∑ i=1mrizi=∑ i=1l r′ivi for some r′i∈R (i=1,…,l). Hence s3s4n=∑ i=1ms4riwi+∑ i=1l r′itvi. Thus s3s4N⊆⟨w1,…,wm,tv1,…,tvl⟩ implies that N is S-finite, which is a contradiction.

Let F=⟨m1,…,mk⟩ be a submodule of M such that sM⊆F for some s∈ S. Claim that 𝔭∩S=∅. Indeed, if s′∈𝔭 for some s'∈ S, then s′M⊆N⊆M. So ss′N⊆ss′M⊆s′F⊆s′M⊆N 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 1≤j≤k. Since mj∈N, 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 s5∈S. Write yi=wi+aimj for some wi∈N and ai∈R (i=1,…,m). Let n ∈ N. Then s5n=∑ i=1mri(wi+aimj)=∑ i=1mriwi+(∑ i=1mriai)mj. Thus (∑ i=1mriai)mj∈N. So ∑ i=1mriai∈𝔭. Thus s5N⊆⟨w1,…,wm⟩+𝔭mj. As Ann(M)⊆(N:M)=𝔭, there exists an S-finite submodule N𝔭 of M such that 𝔭M⊆N𝔭⊆M(𝔭). Thus

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 𝔭M⊆N𝔭⊆M(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

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:M→N is said to be a w-monomorphism (resp., w-epimorphism, w-isomorphism) if for any 𝔭∈w−Max(R), f𝔭:M𝔭→N𝔭 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:F→M. 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 si∈R∖𝔭 such that sirmi∈N. Thus s1⋯snrF⊆N. So s1⋯snrJm⊆N for all m∈M⊆E(M), where E(M) is the injective envelope of M. By [13, Theorem 6.16], s1⋯snrM⊆N since N is a w-module. Hence s1⋯snr∈(N:RM). Consequently, rs=s1⋯snrs1⋯sns∈(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 𝔭M⊆N𝔭⊆M(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 𝔭M⊆N𝔭⊆M(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 N1≤N2 if and only if N1⊆N2 in N. Let {Ni∣i∈Λ} be a chain in N. Set N:=∪ i∈ΛNi. Then N is not w-finite type. Indeed, suppose there is an exact 0→F→N→T→0 with T GV-torsion and F=⟨x1,…,xn⟩ finitely generated. Then there exists i0∈Λ such that F⊆Ni0. 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 N∈N. So by Zorn's Lemma, N has a maximal element, which is also denoted by N. Set 𝔭:=(N:M)={r∈R∣rM⊆N}. 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,b∈R∖𝔭 such that ab∈𝔭. Since a,b∈R∖𝔭, we have aM⊆N and bM⊆N. Therefore N+aM is w-finite type. Let {y1,…,ym} be a subset of N+aM such that 0→F1→N+aM→T1→0 be an exact sequence with T₁ GV-torsion and F1=⟨y1,…,ym⟩ finitely generated. Write yi=wi+azi for some wi∈N and zi∈M (1≤i≤m). Set L:={x∈M∣ax∈N}. Then N+bM⊆L, and hence L is w-finite type. Let 0→F2→L→T2→0 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 J1n⊆F1. So there is {rit∣t=1,…,p;i=1,…,m}⊆R such that

Then ∑ i=1mritzi∈L (t=1,…,p). Thus there exists a GV-ideal J2=⟨j21,…,j2l⟩ such that j2s∑ i=1mritzi=∑ i=1k r′it,sxi for some {r′it,s∣i=1,…,k;t=1,…,p;s=1,…,l}⊆R.

So j1tj2sn=∑ i=1mj2sritwi+∑ i=1k r′it,saxi (t=1,…,k;s=1,…,l). Thus

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 y∈M(𝔭) such that y∈N. Then there exists t′∈R∖𝔭 such that t′y∈𝔭M=(N:M)M⊆N. As t′∈𝔭=(N:M), it follows that t′M⊆N. Therefore N+t'M is w-finite type. Let 0→F3→N+t′M→T3→0 be an exact sequence with T₃ GV-torsion and F3=⟨u1,…,um⟩ a finitely generated submodule of N+t'M. Write ui=wi+t′zi (i=1,…,m) with w_i∈ N and z_i∈ M. Set L:={x∈M∣tx∈N}. Then N⊂N+Ry⊆L, and hence L is w-finite type. Let 0→F4→L→T4→0 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 J3n⊆F3. So there is {rit∣t=1,…,p;i=1,…,m}⊆R such that

So ∑ i=1mritzi∈L (t=1,…,p). Thus there exists a GV-ideal J4=⟨j41,…,j4l⟩ such that j4s∑ i=1mritzi=∑ i=1n r′it,sui for some {r′it,s∣i=1,…,m;t=1,…,p;s=1,…,l}⊆R. So j3tj4sn=∑ i=1mj4sritwi+∑ i=1k r′it,st′ui (t=1,…,k;s=1,…,l). Thus J3J4n⊆⟨w1,…,wm,t′u1,…,t′uk⟩ 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 1≤j≤k. Since mj∈N, it follows that N+Rm_j is w-finite type. Let 0→F5→N+Rmj→T5→0 be an exact sequence with T₅ GV-torsion and F5=⟨y1,…,ym⟩ a finitely generated submodule of N+Rm_j. Write yi=wi+aimj for some w_i∈ N and ai∈R (i=1,…,m). Let n be an element in N. Then there is a GV-ideal J5=⟨j51,…,j5l⟩ such that J5n⊆F5. So there is {rit∣t=1,…,p;i=1,…,m}⊆R such that j5tn=∑ i=1mrityi=∑ i=1mritwi+(∑ i=1mritai)mj (t=1,…,l). So ∑ i=1mritai∈𝔭. Thus J5N⊆⟨w1,…,wm⟩+𝔭mj. As Ann(M)⊆(N:M)=𝔭, there exists a w-finite type submodule N𝔭 of M such that 𝔭M⊆N𝔭⊆M(𝔭). Thus

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 authors would like thank referees for useful comments.

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).