Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 5s (2025) 488 https://internationalpubls.com Class of Modules for Which Strongly Hopfian Modules are Noetherian Mankagna Albert Diompy1, Ousseynou Bousso2, Oumar Diankha3 1,2,3Department of Mathematics and Computer Science, University of Cheikh Anta Diop, Dakar, Senegal Email Id:ousseynou1.bousso@ucad.edu.sn2, oumar.diankha@ucad.edu.sn3 Corresponding author: albertdiompy@yahoo.fr Article History: Received: 13-10-2024 Revised: 28-11-2024 Accepted: 09-12-2024 Abstract: Let R be an arbitrary ring and M a left R-module. In this paper we introduce the modules M such that every strongly Hopfian module in Οƒ[M] is Noetherian. These modules will be called SF-modules. We characterize such modules and study their properties. Relationships between SF-modules and other classes of modules are given. Keywords: Strongly Hopfian module, 𝑆𝐹-module, Perfect module, Locally Noetherian module; Hollow module; Semiartinian module; 𝛱-semiartinian module. AMS subject Classification: 13C05, 13E05, 13F10 1. Introduction The study of modules by properties of their endomorphisms has long been of interest. Throughout in this paper, rings are considered associative, non necessarily commutative with identity 1 β‰  0, all modules are unitary left 𝑅-modules and 𝑅-Mod denotes the category of left unitary 𝑅- modules. We denote by 𝜎[𝑀] the full subcategory of 𝑅-modules whose objects are all 𝑅-Mod subgenerated by 𝑀. A 𝑅-module 𝑀 is Noetherian (resp. artinian) if any ascending (resp. descending) chain of submodules of 𝑀 is stationary. A 𝑅-module 𝑀 is called Hopfian, if any surjective 𝑅-homomorphism 𝑓: 𝑀 β†’ 𝑀 is an isomorphism. An object 𝑁 of 𝜎[M] is said to be strongly Hopfian, if for every 𝑅-endomorphism of 𝑁, the chain πΎπ‘’π‘Ÿπ‘“ βŠ† πΎπ‘’π‘Ÿπ‘“2 βŠ† β‹― βŠ† πΎπ‘’π‘Ÿπ‘“π‘› βŠ† β‹― is stabilizes. A ring 𝑅 is said 𝑆𝐹-ring, if every strongly Hopfian 𝑅-module is Noetherian. Let 𝑅 be a commutative ring, a 𝑅-module 𝑀 is said 𝐹𝐺𝑆- module if every Hopfian object of 𝜎[𝑀] is finitely generated. A 𝑅-module 𝑀 is called endo-noetherian if for any family (𝑓𝑖)𝑖β‰₯1 of endomorphisms of 𝑀, the sequence πΎπ‘’π‘Ÿ(𝑓1) βŠ† πΎπ‘’π‘Ÿ(𝑓2) βŠ† β‹― βŠ† πΎπ‘’π‘Ÿπ‘“π‘› βŠ† β‹― stabilizes. A 𝑅-module 𝑀 is said 𝐸𝐾𝐹𝑁-module if every endo-noetherian object of 𝜎[𝑀] is Noetherian. A ring 𝑅 is said 𝑆-ring, if every Hopfian 𝑅-module is Noetherian and an 𝑅-module 𝑀 is said 𝑆-module if every Hopfian object of 𝜎[𝑀] is Noetherian. An module 𝑀 is hollow, if 𝑀 β‰  0 and submodule of 𝑀 is a small submodule of 𝑀 . All Noetherian module is strongly Hopfian but converse is not always true. For example, the β„€-module 𝑀 = β¨π‘βˆˆπ‘ƒβ„€π‘ is strongly Hopfian but it is not Noetherian, where 𝑃 is the set of all primes. A module is named 𝑆𝐹-module if every strongly object of 𝜎[M] is noetherian In this paper, first we present preliminary results and some fundamental properties of 𝑆𝐹-modules. Secondly we characterize the class of finitely generated and hollow 𝑆𝐹-modules. Additionally, we Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 5s (2025) 489 https://internationalpubls.com prove that in the setting of finitely generated 𝑆𝐹-modules, noetherian module, artinian module and semiartinian module are equivalent. 2. Some properties of SF-modules Lemma 2.1. For a ring R we have : 1. Every Noetherian R-module is endo-noetherian. 2. Every endo-noetherian R-module is strongly Hopfian. 3. Every strongly Hopfian R-module is Hopfian. Proposition 2.2. If M be a SF-module. Then we have the following properties: 1. Every submodule of a strongly Hopfian module in Οƒ[M] is strongly Hopfian. 2. Every quotient of a strongly Hopfian module in Οƒ[M] is strongly Hopfian. Proof. 1) Let N be a submodule of strongly Hopfian module K in Οƒ[M]. As M is an SF-module, then K is Noetherian. Since submodule of Noetherian module is Noetherian so N is Noetherian. Therefore N is strongly Hopfian beacause every Noetherian module is strongly Hopfian. 2) Result from the fact that any quotient of a Noetherian module is Noetherian. β—» Proposition 2.3. Let R be a ring. The following assertions are equivalent: 1. R is SF-ring. 2. Every R-module is a SF-module. Proof. 1) ⟹ 2). Let M a R-module and N a strongly Hopfian objet of Οƒ[M]. Since Οƒ[M] is the full subcategory of R-Mod then N is a strongly Hopfian R-module. As R is a SF-ring then N is Noetherian. 2) ⟹ 1) Suppose that every R-module is SF-module. Let K be a strongly Hofian R-module. Since K ∈ Οƒ[K] then K is Noetherian. Hence R is a SF-ring. β—» Remark 2.4. 1. Every S-module is SF-module. 2. Every SF-module is a EKFN-module. Proposition 2.5. Let R be a commutative ring and M a finitely generated R-module. If M is a SF- module, then every object of Οƒ[M] has a projective cover. Proof. If M is SF-module, then by remark 2.4. M is EKFN-module and by Proposition 3.4. of [5], every object of Οƒ[M] has a projective cover. β—» Proposition 2.6. For a R-module M, the following properties are equivalent: 1. M is an SF-module. 2. Every module in Οƒ[M] is an SF-module. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 5s (2025) 490 https://internationalpubls.com Proof. 1)⟹ 2): Let N ∈ Οƒ[M] then Οƒ[N] is the smallest category of Οƒ[M] containing N and it is a full subcategory of Οƒ[M]. If K is a strongly Hopfian object of Οƒ[N], then K ∈ Οƒ[M] and since M is an SF -module then K is Noetherian. 2)⟹ 1): it’s obvious because M ∈ Οƒ[M]. β—» Proposition 2.7. Over R-artinian ring, all R-module is SF-module. Proof. Let M be a R-module and let N be a strongly Hopfian module in Οƒ[M]. Since R is artinian ring then according to 31.5 of [11], Οƒ[M] = R/Ann(M)-Mod. Hence every module in Οƒ[M] is an R/Ann(M)-module therefore N is a ideal of R/Ann(M). As R is artinian then R/Ann(M) is artinian and so N is finitely generated. Since over artinian ring, finitely generated and Noetherian are equivalent then N is Noetherian. β—» Proposition 2.8. Let M be a R-module. If every module in Οƒ[M] is injective, then M is an SF- module. Proof. Suppose that every module in Οƒ[M] is injective. Let K be a strongly hopfian object of Οƒ[M] then K is hopfian. Since by hypothesis K is injective then according to Theorem 3.5. of [9] , K is Noetherian and therefore M is an SF-module. β—» 3. Characterization of SF-modules Definition 3.1. Let M be an R-module. A module N in Οƒ[M] is semiperfect in Οƒ[M] if every factor module of N has a projective cover in Οƒ[M]. Definition 3.2. N is perfect in Οƒ[M] if, for every index set Ξ›, the sum N (Ξ›) is semiperfect in Οƒ[M]. Lemma 3.3. (see 43.11 in [11]). Let R be a commutative ring, M a finitely generated, self-projective R-module. Then the following statements are equivalent: 1. M perfect in Οƒ[M] 2. οΏ½Μ…οΏ½ = R/An(M) is a perfect ring. Theorem 3.4. Let R be a commutative ring, M a finitely generated, self-projective R-module. Then the following statements are equivalent: 1. M is a SF-module; 2. M perfect in Οƒ[M]; 3. All M-generated flat module in Οƒ[M] is projective in Οƒ[M]. Proof. According to 43.8 in [11], we have the equivalence of assertions (2) and (3). Now let’s prove that 1) is equivalent to 2). 1)⟹ 2): M finitely generated SF-module implies Οƒ[M] = R/An(M)-Mod and M β‰… R/An(M) is an artinian principal ideal ring. Since every artinian ring is perfect ring then M is perfect and so R/An(M) is a perfect ring. Referring to 43.11 in [11], M is perfect in Οƒ[M]. 2)⟹ 1) If M perfect in Οƒ[M] then by 43.11 of [11] , R/An(M) is a perfect ring. Since every perfect Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 5s (2025) 491 https://internationalpubls.com ring is semiperfect and every semiperfect ring is semilocal, then R/Ann(M) is a semilocal ring. By theorem 3.2 in [4], we deduce that R/Ann(M) is an SF-ring. As M is finitely generated Οƒ[M] = R/Ann(M)-Mod and so every module in Οƒ[M] is a R/Ann(M)-module. If N is a strongly Hopfian module in Οƒ[M] then N is Noetherian because R/Ann(M) is an SF-ring and N is a module of R/Ann(M). Therefore M is an SF-module. β—» NB: We denote by Max(M), the set of maximal submodules of a module M. Corollary 3.5. Let R be a commutative ring and M a self-projective hollow module and Max(M) β‰  βˆ…. If M is a SF-module, then S = EndR(M) satisfies the descending chain conditions for cyclic ideals. Proof. Assume M a projective hollow module and Max(M) β‰  βˆ… then according to Theorem 2.2 of [2], M is a finitely generated local module. Then M is finitely generated self-projective. Hence if M is a SF-module then by Theorem 3.4. M perfect in Οƒ[M] and referring to 43.4 of [11], S = EndR(M) satisfies the descending chain conditions for cyclic ideals. β—» Definition 3.6. A module M is called semiartinian if every nonzero homomorphic image of M has nonzero socle. Definition 3.7. A module M is called Ξ -semiartinian if the direct product MI is a semiartinian module for every non empty set I. Definition 3.8. The ring R is called strongly Ο€-regular if for each a ∈ R, there is an integer n β‰₯ 1 and b ∈ R such that an = an+1b. M is called Fitting module if every endomorphism of M satifies Fitting’s lemma (i.e., there exists an integer n β‰₯ 1 such that M = Kerf n βŠ• Imf n ). Theorem 3.9. Let R be a ring and M a finitely generated R-module. If M is SF-module then the following statements are equivalent: 1. M is artinian; 2. M is semiartinian module; 3. Every module in Οƒ[M] is semiartinian ; 4. M is Ξ -semiartinian module; 5. M is Noetherian. Proof. 1)⟹ 2): It’s obvious. 2)⇔ 3): If M is semiartinian module, then by Corollary 2.13. in [8], R/An(M) is a semiartinian ring. Let N an object of Οƒ[M] then since M is finitely generated Οƒ[M] = R/Ann(M)-Mod and therefore N is a module R/Ann(M)-module. It is well know a ring R is semiartinian if and only if every R-module is semiartinian . Since R/Ann(M) is a semiartinian ring then every R/Ann(M)-module is semiartinian and hence N is semiartinian. The converse is trivial. 2)⇔ 4) Result from Corollary 3.3. of [8] Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 5s (2025) 492 https://internationalpubls.com 2)⟹ 5) If M is semiartinian then according to Corollary 2.13. in [8], EndR(M) is a strongly Ο€-regular ring. Therefore by proposition 2.7 of [6], M is Fitting module and so an strongly Hopfian module. Since by hypothesis M is a SF-module so M is Noetherian. 5)⟹ 1) M finitely generated and SF-module implies M β‰… R/An(M) is artinian principal ideal and over artinian principal ideal ring Noetherian module and artinian module coincide. β—» Lemma 3.10. (see proposition 14 of hollow and semihollow modules). Let N be a proper submodule of a module M. If M is a hollow module and M/N is finitely generated, then M is finitely generated. Theorem 3.11. Let R be a commutative ring and M a hollow module. We suppose that for every proper submodule N of M, M/N is finitely generated. Then the following conditions are equivalent: 1. M is a SF-module; 2. M is a locally Noetherian module; 3. M is Noetherian module; Proof. 1)⟹ 2): Let M be a SF-module then by remark 2.4. M is a EKFN-module. By hypothesis, it results from lemma 3.10. that M is finitely generated. Hence by Theorem 3. of [5], M is a locally Noetherian module. 2) ⇔ 3) Result from Corollary 2.3. in [7] Now we prove that 2) ⟹ 1): Let N ∈ Οƒ[M] a strongly Hopfian module. Since M is locally Noetherian then according to Corollary 2.3. in [7], R/Ann(M) is a Noetherian ring. Since M is finitely generated Οƒ[M] = R/Ann(M)-Mod and M β‰… R/Ann(M) is finitely generated and Noetherian. So N ∈ Οƒ[M] implies that N is an ideal of R/Ann(M) and therefore a submodule of M. It’s well know over Noetherian ring, every submodule of finitely generated module is finitely generated. Hence N is Noetherian because over Noetherian ring, finitely generated and Noetherian module coincide. β—» Corollary 3.12. Let R be a commutative ring and M a hollow module. We suppose that for every proper submodule N of M, M/N is finitely generated. Then the following conditions are equivalent: 1. M is a SF-module, 2. Every finitely generated module in Οƒ[M] is Noetherian. 3. Every finitely generated module is finitely presented in Οƒ[M]. 4. Every direct sum of M-injective module in Οƒ[M] is M-injective. Proof. By hypothesis, it results from Lemma 3.10. that, M is finitely generated and according to the theorem 3.11. M is a SF-module if and only if, M is a locally Noetherian module; and referring to 27.3 of [11], we have the result. β—» Theorem 4. Let M be a local R-module, then the following are equivalent: 1. M is a S-module; 2. M is a SF-module; 3. M is of finite length and every submodule of M is cyclic; 4. M is of finite representation type; Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 5s (2025) 493 https://internationalpubls.com 5. M is FGS-module; Proof. 1) ⟹ 2). Result from remark 2.4. 2) ⇔ 3) ⇔ 4) Since M is local SF-module then M is a finite generated SF-module. By Lemma 2.1. and Lemma 3.3. M is isomorphic to R/Ann(M) who is a principal ideal ring. This double equivalence result from Theorem 9 in [10]. 4) ⟹ 5) Result from Theorem 1 in [3]. 5) ⟹ 1) Let N be a Hopfian module in Οƒ[M]. Since M is a FGS-module then N is finite generated. From Proposition 3 in [3] N is Noetherian. Therefore M is a S-module. β—» Acknowledgements The authors would like to express their sincere thanks for the referee for his/her helpful suggestions and comments. Compliance with ethical standards Conflict of interest On behalf of all authors, the corresponding author states that there is no conflict of interest. References [1] F. W. Anderson and K.R. Fuller: Rings and categories of modules, Springer-Verlag, Berlin 1974. [2] A. Azizi: Hollow Modules Over Commutative Rings, Palestine Journal of Mathematics Vol. 3(Spec 1) (2014), 449– 456. [3] A. BA, A. M Diompy, A. Diouf and A. S Diabang: Some Results on FGS -modules, Journal of Mathematics Research; Vol. 9, No. 1; February 2017. [4] M.A Diompy, A.S Diabang, O. Bousso and R.D Diouf: On SF-Rings, International Journal of Algebra, Vol.18, 2024, no 1, 1-10. [5] M.A Diompy, O. Bousso and R.D Diouf: EKFN-Modules, Utilitas Mathematica, 118, 27 - 32 (2024). [6] A. Hmaimou, A. Kaidi and E. Sanchez Campos: Generalized Fitting modules and rings, Journal of Algebra, 308 (2007), 199–214. [7] F. Kourki and R. Tribak: Some results on locally noetherian modules and locally artinian modules. KYUNGPOOK Math. J. 58(2018), 1-8 https://doi.org/10.5666/KMJ.2018.58.1.1 pISSN 1225-6951 eISSN 0454-8124 [8] F. Kourki and R. Tribak: On semiartinian and Ξ -semiartinian modules ,Palestine Journal of Mathematics Vol. 7(Special Issue: I, 2018) , 99–107. [9] F.C Leary: Hopfian and co-Hopfian Modules over Artinian rings, https://arxiv.org/abs/2112.01596v1 (2021). [10] Sangare M. and Kaidi. A: une caracterisation des anneaux artiniens Γ  idΓ©aux principaux. Lect. Note in Math, 328. Springer-Verlag. [11] R. Wisbauer: Foundations of modules and rings theory. Gordon and Breach Science Publishers. (1991).