Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 675 https://internationalpubls.com On Modules For Which Every Cosingular Veri fy The D4- Condition Papa Cheikhou Diop 1 and Modou Seye 1 1Département de Mathématiques, UFR Sciences et Technologies, Université Iba Der Thiam de Thiès, Sénégal Email Id: cheikh.diop@univ-thies.sn, modou.seye@univ-thies.sn Corresponding author: cheikh.diop@univ-thies.sn Article History: Received: 14-11-2024 Revised: 22-12-2024 Accepted: 17-01-2025 Abstract Let R be an associative ring with unity and M an unital left R-module. In this paper we introduce D41-module which is a generalization of D4-module. A module M is called D41 if M = N ⊕ K with N,K ≤ M, K is cosingular and f ∶ N → K is an epimorphism, then ker(𝑓) is a direct summand of N . Some basic properties of these modules are investigated. I t is shown that the class of rings R over which a D41-module is a D4- module is exactly that of COSP -rings. Also, we study the relations between D41-module and other related modules. 2010 Mathematics Subject Classifications: 16D10, 16D40, 16D60, 16L30. Keywords : small submodules, cosingular modules, projectivemodules, D4-modules, D41- modules. 1. Introduction In this paper, we focus on rings that are associative and unitary, and we consider all modules as left modules, unless otherwise specified. We will introduce some important notations. Let M be a module ; we denote N ≤ M to indicate that N is a submodule of M . The notation N ≤ ⊕ M signifies that N is a direct summand of M . For a module M , we denote its injective envelope as E(M). The socle and radical of the module M are represented by Soc(M) and Rad(M), respectively. We use N ⊆ M to indicate that N is a subset of M . We say that M satisfies the following conditions: (D1-Condition): For every submodule N ≤ M , there exists a decomposition M = M1 ⊕ M2 such that M1 ≤ N and N ∩ M2 is small in M . (D2-Condition): For every submodule N ≤ M such that M/N is isomorphic to a direct summand of M , N is also a direct summand of M . (D3-Condition): If M1 and M2 are direct summands of M and M = M1 + M2, then M1 ∩ M2 is a direct summand of M . (D4-Condition): If 𝑀 = 𝑁 ⊕ 𝐾 with 𝑁, 𝐾 ≤ 𝑀 and 𝑓 ∶ 𝑁 → 𝐾 is an epimorphism, then ker(𝑓) ≤⊕ 𝑁. A module that satisfies the Di-condition will be referred to as a Di-module. Every quasi-projective left R-module is a D2-module, every D2-module is a D3-module, and every D3-module is a D4- module. However, there exist examples of D3-modules that are not D2-modules, as well as D2-modules that are not quasi-projective. D1-modules are referred to as lifting modules by Oshiro [16], D2-modules are called direct-projective by W.K. Nicholson in [15], and D3-modules are termed ∩-direct-projective in [2]. mailto:cheikh.diop@univ-thies.sn mailto:modou.seye@univ-thies.sn mailto:cheikh.diop@univ-thies.sn Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 676 https://internationalpubls.com In 2016, N. Ding, Y. Ibrahim, M. Yousif, and Y. Zhou [6] defined a module as a C4-module if, for any submodules A and B of M , when 𝑀 = 𝐴 ⊕ 𝐵 and 𝑓 ∶ 𝐴 → 𝐵 is a homomorphism with ker(𝑓) ≤⊕ 𝐴, then 𝐼𝑚(𝑓) ≤⊕ 𝐵. D4-modules were studied by N. Ding and others in 2017 [4]. Motivated by these developments and the findings in [4] and [6], we introduce the concept of cosingular D4-modules, which generalizes D4-modules and a dual notion of C41-modules [3]. A module is defined as a cosingular D4-module if, for 𝑀 = 𝑁 ⊕ 𝐾 with 𝑁, 𝐾 ≤ 𝑀, K is cosingular and 𝑓 ∶ 𝑁 → 𝐾 as an epimorphism, i t follows that ker(𝑓) ≤⊕ 𝑁. We refer to these as D41-modules. In Section 3, we explore some properties of D41-modules. We demonstrate that every direct summand of a D41-module also inherits this property. We also show that if M ⊕ M is a D41, then M is a D2. It is proved in this section that any D41-module satisfying the SSP satisfies also the SI P . Additionally, we show that if M is a cosingular D41-module, then M/N is also a D41-module and if 𝑀 �̅�(𝑀) is a D41, then M is a D41. It is given some properties of D41-modules related to direct finite modules, square free modules, summand square-free modules, summand dual-quare-free modules, Hopfian modules, ... We begin Section 4 by providing a characterization of D41-modules via COSP -rings. We also show that a ring R is semiregular iff every finitely presented R-module has a D41-cover. 2. Preliminaries Lemma 2.1. Lemma [20, Proposition 2.1] Let 𝑀, 𝑁 and (𝑀𝑖), 𝑖 ∈ 𝐼 be R-modules. Then: a) If 𝑁 ≤ 𝑀, then 𝑍(𝑁) ≤ �̅�(𝑀) and (𝑁+ �̅�(𝑀)) 𝑁 ≤ 𝑍(𝑀 𝑁⁄ ) b) If 𝑓: 𝑀 → 𝑁 is a homomorphism, then 𝑓(𝑍(𝑀)) ≤ 𝑍(𝑁) c) 𝑍 (𝑀 𝑍(𝑀)⁄ ) = 0 d) 𝑍(⊕𝑖∈𝐼 𝑀𝑖) = ⊕𝑖∈𝐼 𝑍(𝑀𝑖) e) 𝑍(∏𝑖∈𝐼𝑀𝑖) ≤ ∏𝑖∈𝐼 𝑍(𝑀𝑖) f) If M = N + S where S is a small module, then 𝑍(𝑀) = 𝑍(𝑁) g) 𝑍(𝑀) is the smallest submodule such that 𝑍 (𝑀 �̅�(𝑀)⁄ ) = 0. Lemma 2.2. [20, Corol lary 2.2] The class of all cosingular modules is closed under submodules, directs sums and directs products. Lemma 2.3. [5, Proposition 2.1] Let M a D3-module such that 𝑀 = 𝐴1 ⊕ 𝐴2 for submodules A1 and A2. 𝐼𝑓 𝑓 ∶ 𝐴1 → 𝐴2 is a homomorphism such that 𝐼𝑚𝑓 ≤⊕ 𝐴2, then 𝑘𝑒𝑟𝑓 ≤⊕ 𝐴1. Proposition 2.4. Let M be a module, N and K two submodules of M such that K is cosingular. Then the following statements are equivalent: 1) If 𝑀 = 𝑁 ⊕ 𝐾 𝑎𝑛𝑑 𝑓 ∶ 𝑁 → 𝐾 an epimorphism, then 𝑘𝑒𝑟𝑓 ≤⊕ 𝑁. 2) if M = N ⊕ K and 𝑓 ∶ 𝑁 → 𝐾 a homomorphism with 𝐼𝑚𝑓 ≤⊕ 𝐾, then 𝑘𝑒𝑟𝑓 ≤⊕ 𝑁. 3) If N ≤ K and 𝑀 𝐾 ≅ 𝑁 ≤⊕ 𝑀 , then 𝐾 ≤⊕ 𝑀. 4) If 𝑀 = 𝑁 + 𝐾, 𝑁 ≤⊕ 𝑀 and 𝑀/𝑁 ≤⊕ 𝑀/𝐾, then 𝑁 ∩ 𝐾 ≤⊕ 𝑀. 5) If N and K are direct summands of M with 𝑀 = 𝑁 + 𝐾 and 𝑀/𝑁 ≅ 𝑀/𝐾, then Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 677 https://internationalpubls.com 𝑁 ∩ 𝐾 ≤⊕ 𝑀 . 6) If 𝑀 = 𝑁 + 𝐾, 𝑁 ≤⊕ 𝑀 and 𝑀/𝑁 ≅ 𝑀/𝐾, then 𝐾 ≤⊕ 𝑀. 7) If 𝑀 = 𝑁 ⊕ 𝑁′ = 𝐾 ⊕ 𝐾′ = 𝑁 + 𝐾 = 𝑁 + 𝐾′, where 𝑁′𝑎𝑛𝑑 𝐾′ are submodules of M, then 𝑁 ∩ 𝐾 ≤⊕ 𝑀 . 8) If N and K are direct summands of M with M = N + K and 𝑁 ≅ 𝐾, then 𝑁 ∩ 𝐾 ≤⊕ 𝑀. Proof: The proof follows by the same method as in [5, Theorem 2.2]. 3. Some properties of D41-modules Throughout this section, we shall investigate some general properties of D41-modules. Definition 3.1. Let M be an R-module. We say that M is a D41 if, 𝑀 = 𝑁 ⊕ 𝐾 for 𝑁, 𝐾 ≤ 𝑀 such that K is cosingular and f : N → K is an epimorphism, then 𝑘𝑒𝑟(𝑓) ≤⊕ 𝑁. The ring R is a right (left) D41-ring if the right R-module RR (left RR) is D41. Example 3.2. 1) Every D4-module has D41. Particularly, each projective module has D41. 2) Each hereditary module is D41 as any submodule of such module is projective. 3) Every semisimple module is a D41-module. 4) Each module with the summand intersection property (SIP ), has D41. Next, we give an example of a D41-module that is not a D4-module. Example 3.3. Consider the module 𝑀 = ℚ ⊕ ℤ, where ℚ is the rational as a ℤ -module and ℤ is the integers as a ℤ -module. We decompose 𝑀 = ℚ ⊕ ℤ. ℤ is cosingular as a ℤ -module. Now, consider an epimorphism 𝑓 ∶ ℚ → ℤ. A natural choice for such an epimorphism is the inclusion map: 𝑓 ∶ ℚ → ℤ, with 𝑓(𝑞) = ⌊𝑞⌋ for 𝑞 ∈ ℚ, where the notation ⌊𝑞⌋ represents the floor function of a real number 𝑞. It is defined as the greatest integer less than or equal to q. Here, the kernel of this map is 𝑘𝑒𝑟(𝑓) = ℚ, which is clearly a direct summand of ℚ since ℚ = ℚ ⊕ {0}. Thus, the D41-condition holds for this map. The 𝐷4-condition requires that for every decomposition 𝑀 = 𝑁 ⊕ 𝐾 and for any epimorphism 𝑓 ∶ 𝑁 → 𝐾, the kernel 𝑘𝑒𝑟(𝑓) must be a direct summand of N, without assuming that K is cosingular. In this case, the map 𝑓 ∶ ℚ → ℤ (in the previous step) satisfies the 𝐷4-condition, because 𝑘𝑒𝑟(𝑓) = ℚ and it is a direct summand of 𝑁 = ℚ . However, if 𝐾 = ℤ which is not cosingular, the kernel might fail to be a direct summand of N. Therefore, the D4-condition does not automatical ly hold unless we impose the cosingularity condition on K , which is present in the D41-condition. Thus, 𝑀 = ℚ ⊕ ℤ is an example of a D41-module that is not a D4-module. The cosingularity of K is what ensures the kernel is a direct summand in the D41-case, but this condition is not guaranteed in the D4-case. Proposition 3.4. Let M be a D41-module. Then any direct summand of M is a D41-module. Proof: Let M be a D41-module and N ≤ ⊕ M . We want to demonstrate that if 𝑁 = 𝐾 ⊕ 𝐾′ such that 𝐾′ is cosingular and 𝑓 ∶ 𝐾 → 𝐾′ is an epimorphism, then 𝑘𝑒𝑟𝑓 ≤⊕ 𝐾. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 678 https://internationalpubls.com Consider 𝑀 = 𝑁 ⊕ 𝑁′ = 𝐾 ⊕ 𝐾′ ⊕ 𝑁′, where 𝑁′ ≤ 𝑀. Define the canonical projection 𝜋 ∶ 𝐾 ⊕ 𝑁′ → 𝐾. It follows that the composition 𝑓 ∘ 𝜋 ∶ 𝐾 ⊕ 𝑁′ → 𝐾′ is also an epimorphism, with 𝑘𝑒𝑟(𝑓 ∘ 𝜋) = 𝑘𝑒𝑟𝑓 ⊕ 𝑁′. Since 𝑀 = (𝐾 ⊕ 𝑁′) ⊕ 𝐾′ is a D41-module, we conclude that 𝑘𝑒𝑟𝑓 ⊕ 𝑁′ ≤⊕ 𝐾 ⊕ 𝑁′ ≤⊕ 𝑀. Therefore, it follows that 𝑘𝑒𝑟𝑓 ≤⊕ 𝑁. Consequently, N is a 𝐷41-module. Proposition 3.5. If 𝑀 ⊕ 𝑀 is a D41-module with M cosingular, then M is a D2-module. Proof: We need to demonstrate that if 𝑁, 𝐾 ≤ 𝑀 with 𝑀 𝐾⁄ ≅ 𝑁 ≤⊕ 𝑀, t he n 𝐾 ≤⊕ 𝑀. To show this, express M as 𝑀 = 𝑁 ⊕ 𝐿 for some L ≤ M . Then we have 𝑀 ⊕ 𝑀 ≅ 𝑀 ⊕ 𝑁 ⊕ 𝐿 ≅ 𝑀 ⊕ (𝑀/𝐾) ⊕ 𝐿. From this, we see that 𝑀 ⊕ (𝑀/𝐾) is a D41-module, which implies that the natural epimorphism M → M /K splits. Consequently, it follows that K ≤ ⊕ M . Proposition 3.6. If M1 ⊕ M2 is a D41-module and there is an epimorphism 𝑓 ∶ 𝑀1 → 𝑀2 with M2 is cosingular, then M2 is a D2-module. Proof: Since 𝑀1 ⊕ 𝑀2 is a D41-module and 𝑓 ∶ 𝑀1 → 𝑀2 is an epimorphism, we have ker f ≤ ⊕ M1. We can express M1 as M1 = N ⊕ ker f , where N ≤ M1. Thus, 𝑁 ≅ 𝑀1 ker(𝑓)⁄ ≅ 𝑀2. Now, we have 𝑀2 ⊕ 𝑀2 ≅ (𝑁 ⊕ 𝑀2) ≤ ⊕ (M1 ⊕ M2). Since M2 ⊕ M2 is a 𝐷41-module (by Proposition 3.4), i t follows that M2 is a D2-module (by Proposition 3.5). Recall that a module M has the summand intersection property (SIP) if the intersection of any two direct summands of M is a direct summand of M . Lemma 3.7. [9, Theorem 1.2] A module M has the summand intersection property (SIP) if and only if, for every decomposition M = A ⊕ B and every 𝛾 ∶ 𝐴 → 𝐵, the kernel of γ is a direct summand of A Corollary 3.8. The fol lowing conditions are equivalent for a D41-module M. 1) M has the SI P property 2) For every decomposition M = N ⊕ N and for every epimorphism 𝑓 ∶ 𝑁 → 𝐾 such that K is cosingular and 𝐼𝑚(𝑓) ≤ ⊕ K , then 𝑘𝑒𝑟(𝑓) ≤ ⊕ N. Proposition 3.9. If every submodule of a D41-module M is a D41-module, then M has the summand intersection property. Proof: Let M be D41-module such that 𝑀 = 𝑁 ⊕ 𝐾 with K cosingular and 𝜆 ∶ 𝑁 → 𝐾. Then 𝑁 ⊕ 𝐼𝑚(𝜆) is a submodule of M . By hypothesis, 𝑁 ⊕ 𝐼𝑚(𝜆) is a D41-module. So, the epimorphism 𝛼 ∶ 𝑁 → 𝐼𝑚(𝜆) splits. Thus 𝑘𝑒𝑟(𝛼) is a direct summand of N . consequently, M has the summand intersection property by lemma 3.7. Definition 3.10. Let M be a left R-module. We say that M satisfies the condition (⋆) if M = N ⊕ K for some submodules N and K of M with K being cosingular, and 𝑓 ∶ 𝑁 → 𝐾 is an R-epimorphism while 𝑖𝑑 ∶ 𝐾 → 𝐾 is the identity map, then there exists an R-homomorphism f ′ : K → N such that f ◦ f ′ = id. It is clear that every D41-module satisfies condition (⋆). Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 679 https://internationalpubls.com Remark 3.11. Given that Morita equivalence preserves summands, epimorphisms, cosingularity, and isomorphisms, it follows that if R and S are Morita-equivalent rings related by a category equivalence ϕ: Mod-R → Mod-S, then a left R-module 𝑀𝑅 satisfies condition (⋆) if and only if ϕ(MR)S also satisfies condition (⋆). Proposition 3.12. Let M ⊕ M be a cosingular D41-module. If M is a dual-Rickart module, then EndR(M) is a Von Neumann regular ring. Proof: Let M ⊕ M be a D41-module and f : M → M an endomorphism. We have 𝑀 ker (𝑓) ≅ 𝐼𝑚(𝑓). Since 𝐼𝑚(𝑓) is a direct summand of M (due to the dual-Rickart condition), we can express M as 𝐼𝑚(𝑓) ⊕ 𝐾 ≅ 𝑀/𝑘𝑒𝑟(𝑓) ⊕ 𝐾 for some K ≤ M . Therefore, we obtain: 𝑀 ⊕ 𝑀 = 𝑀 ⊕ 𝐼𝑚(𝑓) ⊕ 𝐾 ≅ 𝑀 ⊕ 𝑀/𝑘𝑒𝑟(𝑓) ⊕ 𝐾. Since M ⊕ M is a D41-module, i t follows that 𝑀 ⊕ 𝑀/𝑘𝑒𝑟(𝑓) is also a D41-module, which implies that the epimorphism 𝑔 ∶ 𝑀 → 𝑀/𝑘𝑒𝑟(𝑓) splits. As a result, 𝑘𝑒𝑟(𝑓) is a direct summand of M . Hence, EndR(M) is a Von Neumann regular ring. Proposition 3.13. The fol lowing statements are equivalent for a ring R and any finitely generated module M: 1) Every R-module is Rickart ; 2) Every R-module has the SI P property ; 3) Every submodule of a cosingular and finitely generated R-module is a summand ; 4) Every R-module is a D41-module. Proof: 1) ⇒ 2). Every Rickart module has the SI P property (see [12, Proposition 2.16]). 2) ⇒ 3). Let N and K be finitely generated with N ≤ M . By hypothesis, M ⊕ N has the SI P property. Thus, M ∩ N is a direct summand of M ⊕ N , leading to N ≤ ⊕ M . 3) ⇒ 4). Let N and K be direct summands of a finitely generated R-module M with N + K = M and K cosingular. Then, N ∩ K is finitely generated, which implies that N ∩ K ≤ ⊕ M . Hence, M is a D4-module, i.e., a D41-module. 4) ⇒ 1). Let M be a finitely generated R-module and f an endomorphism of M . Since 𝑀 + (𝑀/𝑘𝑒𝑟(𝑓)) is finitely generated and, by hypothesis, is a D41-module, it follows that 𝑘𝑒𝑟(𝑓) is a direct summand of M . Therefore, M is a dual-Rickart module. Recall that a module M has the summand sum property (SSP ) if the sum of any two direct summands of M is also a direct summand of M . Proposition 3.14. Let M be a D41-module. If M possesses the SSP , then M also has the SIP . Proof: Let M be a D41-module with the SSP property. Consider direct summands N and K of M . Since M has the SSP , then N + K is a direct summand of M . By proposition 3.4, 𝑃 = 𝑁 + 𝐾 is a D41-module. Applying the isomorphisms theorems, we have 𝐾 ≅ 𝑃 𝑁 ≅ 𝑁+𝐾 𝑁 ≅ 𝑁/𝑁 ∩ 𝐾 . Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 680 https://internationalpubls.com Since P is a D41-module, then the epimorphism 𝑓 ∶ 𝐾 → 𝑁/(𝑁 ∩ 𝐾) splits. So, N ∩ K ≤ ⊕ K and hence a direct summand of M proving that M has the SIP . Proposition 3.15. Let M be a module such that 𝑀/𝑍(𝑀) is a D41-module. Then M is a D41-module. Proof: Let N be a cosingular submodule of M containing 𝑍(𝑀) such that 𝑀 𝑁 ≅ 𝐾, for 𝐾 a direct summand of 𝑀 with 𝐾 ≤ 𝑁. Since 𝑀/𝑍(𝑀) is a D41-module, there exists a direct summand 𝐿 of 𝑀/𝑍(𝑀) such that ( 𝑀 �̅�(𝑀) ) ( 𝑁 �̅�(𝑀) ) ⁄ ≅ 𝐿 and 𝐿 ≤ 𝑁 �̅�(𝑀) . Applying the isomorphism theorem, we have 𝑀 𝑁 ≅ ( 𝑀 �̅�(𝑀) ) ( 𝑁 �̅�(𝑀) ) ⁄ ≅ 𝐿 ≅ 𝐾 ≤⊕ 𝑀. As 𝑁 �̅�(𝑀) ≤⊕ 𝑀/𝑁, thus 𝑁 ≤⊕ 𝑀. Hence, 𝑀 is a D41-module. The converse of the previous proposition holds (see 3.17). Corollary 3.16. Let M = M1 ⊕ M2 a direct sum of submodules M1 and M2 such that �̅�(𝑀1) = 𝑀1 and M2 a cosingular D41-module. Then M is a D41-module. Proof: Since �̅�(𝑀2) = 0, then �̅�(𝑀) = 𝑀1. Thus, 𝑀 �̅�(𝑀) ≅ 𝑀2 which is a D41-module. Hence M is a D41-module by the proposition 3.15. Proposition 3.17. Let M be a D41-module. Then, for every submodule N of M, M/N is a D41-module. Proof: Let M be a D41-module and N a submodule of M and let K = P /N be a co-singular submodule of M/N with P a submodule of M containing N such that (𝑀/𝑁)/(𝑃/𝑁) ≅ 𝐿, L ≤ ⊕ M/N and L ≤ P /N . By the second theorem of isomorphism, we have: 𝑀/𝑃 ≅ (𝑀/𝑁)/(𝑃/𝑁) ≅ 𝐿. As M is a D41-module, then P is a direct summand of M . Hence K = P /N is a direct summand of M/N . Thus M/N is a D41-module. Recall that a module is called directly finite if it is not isomorphic to a proper summand of itself. A module is called square-free if it contains no nonzero submodules isomorphic to a square 𝑁 ⊕ 𝑁. Proposition 3.18. Let M = N ⊕ K be a D41-module, with epimorphisms 𝑓 ∶ 𝑁 → 𝐾 𝑎𝑛𝑑 ℎ ∶ 𝐾 → 𝑁 such that 𝑘𝑒𝑟(𝑓) is cosingular. 1) If K is cosingular and directly finite, then 𝑁 ≅ 𝐾. 2) If N is square-free and K is cosingular, then 𝑁 ≅ 𝐾. Proof: 1) Assume K is directly finite and cosingular. Since M is a D41-module, the epi-morphism f splits, allowing us to express K as 𝐾 ≅ 𝐿, where L ≤ ⊕ N . Write N = L ⊕ K ′ for some submodule K ′ ≤ M . Since direct summands of a D41-module remain D41-modules, L ⊕ K is also a D41-module. 𝐿𝑒𝑡 𝜋 ∶ 𝑁 → 𝐿 be the natural projection. The composition 𝜋 ∘ ℎ ∶ 𝐾 → 𝐿 is an epimorphism that splits by Lemma 2.4. Therefore, we can express K as 𝐾 = 𝑘𝑒𝑟(𝜋 ∘ ℎ) ⊕ 𝑃 for some submodule P ≤ M . Given 𝐾 ≅ 𝐿 ≅ 𝐾/𝑘𝑒𝑟(𝜋 ∘ ℎ) ≅ 𝑃 and that K is directly finite, we have 𝑘𝑒𝑟(𝜋 ∘ ℎ) = 0. Thus, h is an isomorphism since 𝑘𝑒𝑟ℎ ≤ 𝑘𝑒𝑟(𝜋 ∘ ℎ), leading to the conclusion that 𝑁 ≅ 𝐾. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 681 https://internationalpubls.com = 2 ) Assume N is square-free. Since M is a D41-module, the epimorphism f splits, allowing us to write N = ker f ⊕ L for a submodule L isomorphic to K . Consider the epimorphism 𝜋 ∘ ℎ ∶ 𝐾 → 𝑘𝑒𝑟𝑓 with the natural projection 𝜋 ∶ 𝑁 → 𝑘𝑒𝑟𝑓. The direct summands of D41-modules remain D41- modules, so ker f ⊕ K is a D41-module. By Lemma 2.4, the epimorphism 𝜋 ∘ ℎ also splits, yielding 𝐾 = 𝑘𝑒𝑟(𝜋 ∘ ℎ) ⊕ 𝑄 for some 𝑄 ≤ 𝑀 with 𝑘𝑒𝑟𝑓 ≅ 𝑄. Let 𝜙 ∶ 𝐾 → 𝑄 be the natural projection. As a direct summand of M , 𝐿 ⊕ 𝑄 is a D41-module, so the epimorphism 𝜙 ∘ 𝜓 ∶ 𝐿 → 𝑄 splits by Lemma 2.4, where 𝜓 ∶ 𝐿 → 𝐾 is an isomorphism. Hence, we have 𝐿 = 𝑘𝑒𝑟(𝜙 ∘ 𝜓) ⊕ 𝐿′ where 𝑄 ≅ 𝐿′ ≤ 𝑀. This leads to the expression, 𝑁 = 𝑘𝑒𝑟𝑓 ⊕ 𝐿 = 𝑘𝑒𝑟𝑓 ⊕ 𝑘𝑒𝑟(𝜙 ∘ 𝜓) ⊕ 𝐿′ with 𝐿′ ≅ 𝑄 ≅ 𝑘𝑒𝑟𝑓. Since N is square-free, i t follows that 𝑘𝑒𝑟𝑓 = 0, thus 𝑁 ≅ 𝐾. Recall that two direct summands N and K of a module M are called perspective in [13] exactly when they have a common (direct sum) complement L, i.e. M = N ⊕ L = K ⊕ L . Proposition 3.19. The following conditions for a module M are equivalent: 1) M is a D41-module. 2) If N and K are perspective direct summands of M with common direct sum complement is cosingular and N + K = M, then N ∩ K is a direct summand of M. 3) If N and K are perspective direct summands of M with common direct sum complement is cosingular and N + K is a direct summand of M, then N ∩ K is also a direct summand of M. Proof: 1) ⇒ 2) Let N and K be perspective direct summands with a common direct sum complement L and L is cosingular, and suppose N + K = M . Define the projection 𝜋 ∶ 𝑀 → 𝐿 with 𝑘𝑒𝑟𝜋 = 𝐾. Consider the restricted map 𝜋|𝑁 ∶ 𝑁 → 𝐿. Since 𝑁 + 𝐾 = 𝑀, i t follows that π(N) = L ≤ ⊕ L, which implies 𝑘𝑒𝑟𝜋|𝑁 ≤ ⊕ N . Thus, since 𝑘𝑒𝑟𝜋|𝑁 = 𝑁 ∩ 𝐾 ≤ ⊕ N ≤ ⊕ M , we have N ∩ K ≤ ⊕ M . 2) ⇒ 3) Let N and K be perspective direct summands of M such that N + K ≤ ⊕ M . There exist L and P ≤ ⊕ M such that M = N ⊕ L = K ⊕ L = (N + K ) ⊕ P with L is cosingular. By modularity, we can express N + K = N ⊕ (L ∩ (N + K )) and N + K = K ⊕ (L ∩ (N + K )). The modules N ⊕ P and K ⊕ P are then perspective direct summands of M , satisfying (N ⊕ P ) + (K ⊕ P ) = M . By the hypothesis, this implies that (N ⊕ P ) ∩ (K ⊕ P ) = (N ∩ K ) ⊕ P ≤ ⊕ M . Thus, we conclude that N ∩ K ≤ ⊕ M . 3) ⇒ 1) Let M = N ⊕ K and let 𝑓 ∶ 𝑁 → 𝐾 be an epimorphism with K is cosingular. Define the graph submodule 𝐺 = {𝑎 + 𝑓(𝑎) ∶ 𝑎 ∈ 𝐴} within M . Clearly, M = G + K and G ∩ K = 0. This shows that M = G ⊕ K = N ⊕ K , meaning N and G are perspective direct summands of M . Since f is an epimorphism, we have N + G = M . By the hypothesis, i t follows that N ∩ G ≤ ⊕ M . It can be readily demonstrated that N ∩ G = ker f , leading to the conclusion that ker f ≤ ⊕ M . Therefore, we find that ker f ≤ ⊕ N . Corollary 3.20. The following conditions for a module M are equivalent: 1) M is a D41-module. 2) If M = N + K for any perspective direct summands N and K of M with common direct sum complement is cosingular, then there exists a submodule K ′ ⊆ K such that M = N ⊕ K ′ . Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 682 https://internationalpubls.com Proof: 1) ⇒ 2) Let M = N + K , where N and K are perspective direct summands of M with common direct sum complement is cosingular. By Proposition 3.19, we can express M as M = (N ∩ K ) ⊕ L for some submodule L. According to the modular law, we have K = (N ∩ K ) ⊕ (L ∩ K ). Thus, we can write M = N ⊕ (L ∩ K ), leading us to define K ′ := L ∩ K . 2) ⇒ 1) Now let M = N + K , where N and K are perspective direct summands of M with common direct sum complement is cosingular. By the assumption, there exists a submodule 𝐾′ ⊆ 𝐾 such that M = N ⊕ K ′ . This means we can express K as K = (N ∩ K ) ⊕ K ′ . Consequently, N ∩ K is a direct summand of K , and therefore also a direct summand of M . Recall that a module M is a C4-module If M = N ⊕ K , then every monomorphism f : N → K splits [6]. We next give a relationship between C4-modules and D41-modules under certain conditions and having perspective direct summand. A module M is referred to as a DSF-module (dual square-free module) if i t does not contain any proper submodules N and K such that M = N + K and M/N ≅ M / K [4]. Additionally, M is called summand-dual-square-free (SDSF) if the submodules N and K are direct summands of M . Proposition 3.21. The following conditions on a module M are equivalent: 1) M is a D41- and summand-square-free module. 2) M is a C4- and summand-dual-square-free module. Proof: 1) ⇒ 2) Clearly, M is a C4-module by [6]. Next, we show that M is summand-dual-square-free. Suppose M is not summand-dual-square-free ; then there exist two non-zero proper direct summands N and K of M such that N +K = M with K cosingular and M/N ≅ M /K . Since M is a D41-module, we have N ∩ K ≤ ⊕ M . We can write M = (N ∩ K ) ⊕ L, leading to N = (N ∩ K ) ⊕ (N ∩ L) and K = (N ∩ K ) ⊕ (K ∩ L). This gives us N ∩ L ≅ N/(N ∩ K ) ≅ M / K ≅ M/N ≅ K/(N ∩ K ) ≅ K ∩ L with (N ∩ L) ∩ (K ∩ L) = (N ∩ K ) ∩ L = 0, meaning both N ∩ L and K ∩ L are direct summands of M . Since M is summand-square-free, we have N ∩ L = K ∩ L = 0. Thus, N = (N ∩ K ) = K , leading to the contradiction M = N + K = N = K . Hence, M is summand-dual-square-free. 2 ) ⇒ 1) Clearly, M is a D41-module by [4]. Now, we demonstrate that M is summand-square- free. Assume M is not summand-square-free, and let N and K be non-zero direct summands of M with N ≅ K and N ∩ K = 0. Since M is a C4-module, we have N ⊕ K ≤ ⊕ M . We can write M = N ⊕ K ⊕ L for some submodule L ≤ M . Now, we have M/(N ⊕ L) ≅ K ≅ N ≅ M /(K ⊕ L) with 𝑀 = 𝑁 ⊕ 𝐾 ⊕ 𝐿 = (𝑁 ⊕ 𝐿) + (𝐾 ⊕ 𝐿), where both 𝑁 ⊕ 𝐿 and 𝐾 ⊕ 𝐿 are direct summands of M . Since M is summand-dual-square-free, it follows that 𝑀 = 𝑁 ⊕ 𝐿 = 𝐾 ⊕ 𝐿, implying N = K = 0, which leads to a contradiction. Therefore, M is summand-square-free. Proposition 3.22. Let 𝑀 =⊕𝑖∈𝐼 𝑀𝑖 be a direct sum of submodules 𝑀𝑖 . If for every submodule N of M, we have 𝑁 =⊕𝑖∈𝐼 (𝑁 ∩ 𝑀𝑖 ), then M is a D41-module if and only if each 𝑀 𝑖 (for i ∈ I ) is also a D41- module. Proof: Assume that each 𝑀𝑖 is a D41-module for every i ∈ I. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 683 https://internationalpubls.com L L Let M = N ⊕ L = K ⊕ L with N + K = M and K cosingular. According to the hypothesis, we can express 𝑁 =⊕𝑖∈𝐼 (𝑁 ∩ 𝑀𝑖 ) , 𝐾 =⊕𝑖∈𝐼 (𝐾 ∩ 𝑀𝑖 ) and 𝐿 =⊕𝑖∈𝐼 (𝐿 ∩ 𝑀𝑖 ) Since M = N ⊕ L = K ⊕ L, we have: 𝑀 = ⊕𝑖∈𝐼 [(𝑁 ∩ 𝑀𝑖 ) ⊕ (𝐿 ∩ 𝑀𝑖 ) ] =⊕𝑖∈𝐼 [(𝐾 ∩ 𝑀𝑖 ) ⊕ (𝐿 ∩ 𝑀𝑖 ) ] Thus, i t follows that Mi = (N ∩ Mi) ⊕ (L ∩ Mi) = (K ∩ Mi) ⊕ (L ∩ Mi) for every i ∈ I . Additionally, since M = N + K , we have : 𝑀 =⊕𝑖∈𝐼 [(𝑁 ∩ 𝑀𝑖 ) + (𝐾 ∩ 𝑀𝑖 ) ] , which implies Mi = (N ∩ Mi) + (K ∩ Mi). Given that N ∩ Mi and K ∩ Mi are perspective direct summands of Mi with (N ∩ Mi) + (K ∩ Mi) = Mi , i t follows that (N ∩ Mi) ∩ (K ∩ Mi) ≤ ⊕ Mi for every i ∈ I . Consequently, we have N ∩ K = ⊕𝑖∈𝐼 (𝑁 ∩ 𝑀𝑖 ) ∩ ⊕𝑖∈𝐼 (𝐾 ∩ 𝑀𝑖 ) = ⊕𝑖∈𝐼 [(N ∩ Mi) ∩ (K ∩ Mi)] ≤ ⊕ M, which demonstrates that M is a D41-module. The converse is straightforward, as a direct summand of a D41-module is itself a D41-module by proposition 3.4. Recall that a module M is called Hopfian if every surjective R-homomorphism 𝑓 ∶ 𝑀 → 𝑀 is an automorphism. Definition 3.23. An R-module M is termed cosingular Hopfian if every cosingular epimorphism 𝜉 ∶ 𝑀 → 𝑀 is an automorphism. Proposition 3.24. Every indecomposable cosingular Hopfian module is a D41-module. Proof: Let M be an indecomposable cosingular Hopfian left R-module, and let 𝑓 ∶ 𝑀 → 𝑁 be a cosingular epimorphism where N is a direct summand of M . In this case, N must either be 0 or M . In the first scenario, f trivially splits. In the second scenario, where N = M , f is an automorphism and thus also splits. A left R-module M is said to be generalized Hopfian if every surjective R-endomorphism f of M is superfluous, meaning 𝑘𝑒𝑟(𝑓) ≪ 𝑀. Proposition 3.25. Every generalized Hopfian D41-module has a D41-cover. Proof: The proof is straightforward. Proposition 3.26. Let M be a lifting module and consider the following conditions: 1) M is Hopfian ; 2) M is generalized Hopfian ; 3) M is Dedekind finite. Then, 1) ⇒ 2) ⇒ 3) (by [8, Corollary 1.4]). If in addition, M ⊕ M is a D41-module, then we have the equivalence. Proof: For 3) ⇒ 1): Since M ⊕ M is a D41-module, then M is a D2-module. If 𝑓 ∶ 𝑀 → 𝑀 is an epimorphism, then there exists an endomorphism g of M such that 𝑓𝑔 = 1. Given that End(M) is a directly finite ring, we can conclude that 𝑔𝑓 = 1. Therefore, f is an isomorphism, which implies that M is Hopfian. In this section we introduce the concept of t-cosingular D4-module. The notion of t-cosingular module was introduced by Y. Talebi and A. R. M. Hamzekolaee in 2013 (see [17]). A module M is Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 684 https://internationalpubls.com called t-cosingular if 𝑍𝑡̅̅ ̅(𝑀) = 0 where 𝑍𝑡̅̅ ̅(𝑀) = 𝑅𝑒𝑗(𝑀, 𝑇𝑆) = {ker f | f : M → L, L ∈ TS} with TS the class of t-small modules. Definition 3.27. Let M be an R-module. We say that M is a t-cosingular D4-module if, M = N ⊕ K for N, K ≤ M such that K is t-cosingular and f : N → K is an epimorphism, then 𝑘𝑒𝑟(𝑓) ≤ ⊕ N. The ring R is a right (left) t-cosingular D4-ring if the right R-module RR (left RR) is t- cosingular D4. Remark 3.28. It has been proved in [17, Remark 1] that 𝑍𝑡̅̅ ̅(𝑀) ⊆ �̅�(𝑀). From this, we can say that every D41-module is also a t-cosingular D4-module. Proposition 3.29. Any direct summand of a t-cosingular D4-module is again a t-cosingular D4- module. Proof: Let M be a t-cosingular D4-module, and suppose N ≤ ⊕ M . We aim to show that if N = K ⊕ K ′ , where K ′ is t-cosingular and f : K → K ′ is an epimorphism, then ker f ≤ ⊕ K . Assume 𝑀 = 𝑁 ⊕ 𝑁′ = 𝐾 ⊕ 𝐾′ ⊕ 𝑁′, where 𝑁′ ≤ 𝑀. Define the canonical projection 𝜋 ∶ 𝐾 ⊕ 𝑁′ → 𝐾. Then the composition 𝑓 ∘ 𝜋 ∶ 𝐾 ⊕ 𝑁′ → 𝐾′ is an epimorphism, and its kernel is given by 𝑘𝑒𝑟(𝑓 ∘ 𝜋) = 𝑘𝑒𝑟𝑓 ⊕ 𝑁′. Since 𝑀 = (𝐾 ⊕ 𝑁′) ⊕ 𝐾′ is a t-cosingular D4-module, we deduce that 𝑘𝑒𝑟𝑓 ⊕ N ′ ≤ ⊕ K ⊕ N ′ ≤ ⊕ M . Thus, ker f ≤ ⊕ N . Consequently, N is a t-cosingular D4- module. Proposition 3.30. Let M be a t-cosingular D4-module. Then, for every submodule N of M, the quotient module M/N is also a t-cosingular D4-module. Proof: Let M be a t-cosingular D4-module, and let N be a submodule of M . Consider a t-cosingular submodule K = P /N of M/N , where P is a submodule of M containing N . Assume that (M/N )/(P /N ) ≅ L, where L is a direct summand of M/N and L ≤ P /N . By the second isomorphism theorem, we have: M/P ≅(M/N )/(P /N ) ≅ L. Since M is a t-cosingular D4-module, P is a direct summand of M . Consequently, K = P /N is a direct summand of M/N . Therefore, M/N is a t-cosingular D4-module. Recall that a module M has C∗ if every submodule N of M contains a direct summand K of M such that N / K is cosingular [19] and a module M is lifting if every submodule N of M contains a direct summand K of M such that N / K ≪ M/K . We now define a module having t-C∗ if every submodule N of M contains a direct summand K of M such that N / K is t-cosingular. We recall a module is t- lifting if every submodule N of M contains a direct summand K of M such that N / K is t-small in M /K . Proposition 3.31. Every t-cosingular module (and consequently, every t-small module) satisfies the property t-C∗. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 685 https://internationalpubls.com Proof: Since a submodule of a t-cosingular module is t-cosingular, the Proposition follows easily. Proposition 3.32. For an R-module M, the fol lowing statements are equivalent: 1) M satisfies t-C∗. 2) For every submodule N of M, there exists a decomposition M = M1 ⊕ M2 such that M1 ≤ N and N ∩ M2 is t-cosingular. 3) For every submodule N of M, N can be decomposed as N = N1 ⊕ N2 , where N1 is a direct summand of M, and N2 is t-cosingular. Proof: 1) ⇒ 2) Let N ≤ M . By definition, there exists a decomposition M = M1 ⊕ M2 such that M1 ≤ N and N/M1 is t-cosingular. This gives N = M1 ⊕ (N ∩ M2), and since N ∩ M2 ≅ N/M1, it follows that N ∩ M2 is t-cosingular. 2) ⇒ 3) Assume M = M1 ⊕ M2 with M1 ≤ N . Then N = M1 ⊕ (N ∩M2). Define N1 = M1 and N2 = N ∩ M2, satisfying the required decomposition. 3) ⇒ 1) Suppose N ≤ M . By assumption, N = N1 ⊕ N2, where N1 is a direct summand of M , and N2 is t-cosingular. Since 𝑁/𝑁1 ≅ 𝑁2 and N2 is t-cosingular, N/N1 is also t-cosingular. Hence, M satisfies t-C∗ . Proposition 3.33. The following statements are equivalent for a ring R: 1) Every right R-module satisfies t-C∗. 2) Every injective right R-module satisfies t-C∗. 3) Every right R-module can be expressed as a direct sum of an injective module and a t-cosingular module. Proof: 1) ⇔ 2) This equivalence is evident, as any submodule of a module that satisfies t-C∗ also satisfies t- C∗ . 2) ⇒ 3) Let M be an injective module that satisfies t-C∗. By Proposition 3.32, every submodule of M can be expressed as a direct sum of an injective module and a t-cosingular module. 3) ⇒ 1) If every submodule of M can be expressed as a direct sum of an injective module and a t- cosingular module, then M satisfies t-C∗. This follows from the fact that injective submodules are direct summands, as established in Proposition 3.32. Proposition 3.34. Let M = M1 ⊕ M2, where M1 is semisimple and M2 satisfies t-C∗. Then M also satisfies t-C∗. Proof: Let M = M1 ⊕ M2, where M1 is semisimple and M2 satisfies t-C∗ . Let N ≤ M . Then M1 can be decomposed as M1 = (N ∩ M1) ⊕ M ′ for some submodule M ′ ≤ M1. Consequently, M = (N ∩ M1) ⊕ M ′ ⊕ M2 and N = (N ∩ M1) ⊕ A, where A = N ∩ (M ′ ⊕ M2). Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 686 https://internationalpubls.com Since (M2 ⊕ M ′ ) /M ′ satisfies t-C∗, i t follows that (A + M ′)/M ′ = K/M ′ ⊕ L/M ′ for some submodules K and L containing M ′, where K/M ′ is a direct summand of (M2 ⊕ M ′ ) /M ′ and L/M ′ is t-cosingular. Thus, K is a direct summand of M . Moreover, since K = M ′ ⊕ (K ∩ A), K ∩ A is also a direct summand of M . Therefore, (N ∩ M ⊕ (K ∩ A) is a direct summand of M . Finally, N / ( (N ∩ M1) ⊕ (K ∩ A))≅ A/(K ∩ A) ≅ (A + K ) /K ≅ (A + M ′) /K≅ L/M ′ is t- cosingular. Hence, M satisfies t-C∗. Proposition 3.35. Every t-lifting module satisfies the t-C∗ property. Proof It’s obvious. 4. Rings over which certain modules have D41 In this section, we provide a characterization of certain classes of rings in relation to D41-modules. We start with a characterization of the class of rings R for which every D41-module is also a D4-module. Recall that a ring is cosingular projective (COSP ) if every cosingular module over R is projective (see [18, 3]). Theorem 4.1. The following conditions are equivalent for a semisimple ring R: 1) R is a left cosingular projective-ring ; 2) Every D41-R-module is a D4-module ; Proof: 1) ⇒ 2) it’s obvious. 2) ⇒ 1) Since R is a semisimple ring then every module of R is projective. In particular every cosingular module is projective. Thus, R is a cosingular projective-ring. Remark 4.2. Let R be a ring which is not a left cosingular projective-ring. From Theorem 4.1, it follows that R has a D41-module that is not a cosingular projective-module. Theorem 4.3. The following statements are equivalent: 1) R is a left cosingular semisimple ring. 2) Every left R-module is cosingular projective. 3) Every left R-module is cosingular direct-projective. 4) Every left R-module is a D41-module. 5) Every submodule of a D41-module module is a D41-module. 6) Every direct sum of D41-modules is a D41-module. 7) Every direct sum of two cyclic D41-modules is a D41-module. Proof: The implications 1) ⇒ 2) ⇒ 3) ⇒ 4) ⇒ 5) ⇒ 6) ⇒ 7) are obvious. 6) ⇒ 1) Let V be a simple R-module and 𝑓 ∶ 𝑅 → 𝑉 an R-epimorphism. Since R ⊕ V is a D41- module, by Proposition 2.4, V is isomorphic to a direct summand of R, implying that V is projective. Therefore, R is semisimple. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 687 https://internationalpubls.com A module M is termed regular if every cyclic submodule of M is a direct summand of M . This is equivalent to stating that every finitely generated submodule of M is also a direct summand of M (see [21, p. 67]). According to [11], a module M is defined as d-Rickart if for every endomorphism φ of M , the image 𝐼𝑚(𝜑) is a direct summand of M . Next, we provide a characterization in terms of D41-modules for a left semi-hereditary ring to be von Neumann regular. Proposition 4.4. The fol lowing conditions are equivalent for a left semi-hereditary ring R: 1) Every finitely generated R-module is a D4-module ; 2) Every finitely generated R-module is a D41-module ; 3) Every finitely generated projective R-module is a d-Rickart module ; 4) Every finitely generated projective R-module is a regular module ; 5) R is a von Neumann regular ring. Proof: This is immediate. Proposition 4.5. Let A be a class of cosingular R-module and closed under iso-morphisms and directs summand. The following conditions are equivalent: 1) All 𝐴 ∈ 𝐴 is A-projective. 2) Every left R-module in A is a D2-module. 3) Every left R-module in A is a cosingular direct-projective module. 4) Every left R-module in A is a D4module. 5) Every left R-module in A satisfies (D41). Proof: The implications 1) ⇒ 2) ⇒ 3) ⇒ 4) ⇒ 5) are obvious. The implication 5) ⇒ 1) is demonstrated in [14, Proposition 3.6], and we restate it here for clarity. Let N ∈ A and consider the epimorphism R(I ) → N . This implies that R(I ) ⊕ N is an A-D41 module. As noted in [14, Proposition 2.6], N is isomorphic to a direct summand of R(I ). Therefore, N is a projective module. Recall that a submodule A of an R-module B is termed a pure submodule if, for any left R-module X , the natural homomorphism A ⊗ X → B ⊗ X is injective. A module M is called pure-injective if every homomorphism from a pure submodule A of any module B can be extended to a homomorphism from B into M . It is established that both direct summands and direct products of pure-injective modules are also pure-injective. Proposition 4.6. The following conditions are equivalent for a ring R : 1) R is a semisimple Artinian ring. 2) Every R-module is a D41-module. 3) Every direct sum of two cyclic D41-modules is a D41-module. 4) Every 2-generated R-module is a D41-module. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 688 https://internationalpubls.com 5) Every factor module of an injective R-module is a D41-module. 6) R is a von Neumann regular ring in which every injective module is a D41-module. 7) Every pure-injective R-module is a D41-module. Proof: 1) ⇒ 𝑖) for 𝑖 = 2, . . . ,7 is straightforward. The implications 2) ⇒ 3) and 4) ⇒ 3) are also clear. To show 3) ⇒ 1), let S be a simple R-module and consider the R-epimorphism 𝑓 ∶ 𝑅 → 𝑆. Since R ⊕ S is a D41-module, Lemma 2.4 implies that S is isomorphic to a direct summand of R, which indicates that S is projective. Thus, R must be semisimple Artinian. For 5) ⇒ 1), if N is a right ideal of R, then 𝐸(𝑅) ⊕ 𝐸(𝑅)/𝑁 is a D41-module by 5). Therefore, the canonical homomorphism 𝜂 ∶ 𝐸(𝑅) → 𝐸(𝑅)/𝑁 splits by Lemma 2.4, which means that N is a direct summand of E(R) and hence a direct summand of R. This shows that R is semisimple Artinian. Now, for 6 ⟹ 1), it suffices to demonstrate that R is right Noetherian. For a countable family of injective R-modules {𝑀𝑖: 𝑖 = 1, 2, … }, the module (∏ 𝑀𝑖 )∞ 𝑖=1 /⊕∞ 𝑖=1 𝑀𝑖 is pure-injective by [7, Theorem 38.1 and Corollary 42.2]. Since R is Von Neumann regular, this module is also injective by [22, 37.6]. Therefore, by 6), (∏ 𝑀𝑖 )∞ 𝑖=1 ⊕ [ ∏ 𝑀𝑖 ∞ 𝑖=1 ⊕∞ 𝑖=1𝑀𝑖 ] is a D41-module. This implies that the natural epimorphism ∏ 𝑀𝑖 ∞ 𝑖=1 → ∏ 𝑀𝑖 ∞ 𝑖=1 ⊕∞ 𝑖=1𝑀𝑖 splits, leading to ⊕∞ 𝑖=1 𝑀𝑖 ≤⊕ ∏ 𝑀𝑖 ∞ 𝑖=1 . Thus ⊕∞ 𝑖=1 𝑀𝑖 is injective, demonstrating that R is right Noetherian. Finally, for 7) ⟹ 2), consider a countable family of pure-injective R-modules {𝑀𝑖: 𝑖 = 1, 2, … }. The module (∏ 𝑀𝑖 )∞ 𝑖=1 /⊕∞ 𝑖=1 𝑀𝑖 is pure-injective by [7, Theorem 38.1 and Corollary 42.2], and hence it is a D41-module according to 7). Consequently, the natural epimorphism ∏ 𝑀𝑖 ∞ 𝑖=1 → ∏ 𝑀𝑖 ∞ 𝑖=1 ⊕∞ 𝑖=1𝑀𝑖 splits, leading to ⊕∞ 𝑖=1 𝑀𝑖 ≤⊕ ∏ 𝑀𝑖 ∞ 𝑖=1 . Therefore ⊕∞ 𝑖=1 𝑀𝑖 is pure-injective [22, 53.7], this means every R-modules is pure-injective, thus every R-module is a D41-module by 7). Recall that a ring is called perfect if R/Rad(R) is semisimple and Rad(R) is T -nilpotent (see [10, Definition 23.18]). An R-module M is termed finitely presented (or finitely related) if there exists an exact sequence 0 → 𝐾 → 𝐹 → 𝑀 → 0 of R-modules, where F is a free module and both F and K are finitely generated. A ring R is referred to as semiregular if every finitely presented R-module possesses a projective cover (see [15, Theorem 2.9]). Proposition 4.7. The following statements are equivalent for a ring R: 1) R is semiregular. 2) Every finitely presented R-module has a D41-cover. Proof: 1) ⇒ 2): This is evident. 1) ⇒ 2): Let M be a finitely presented R-module and 𝑔: 𝐹 → 𝑀 be a homomorphism from a free R-module F to M . According to 2), F ⊕ M has a D41-cover, and by [4, Theorem 3.2], M has a projective cover. Therefore, R is a semiregular ring. Proposition 4.8. The following statements are equivalent: 1) R is left perfect ; 2) Every flat left R-module is quasi-projective ; Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 689 https://internationalpubls.com 3) Every flat left R-module is a D41-module ; 4) Every left R-module has a D41-cover. 5) Every countably generated R-module has a D41-cover. Proof: The implications 1) ⇒ 2) ⇒ 3) and 1) ⇒ 4) ⇒ 5) are straightforward. 3) ⇒ 1). Let M be a flat left R-module, N a free left R-module, and 𝑔 ∶ 𝑁 → 𝑀 an R-epimorphism. Since K = M ⊕ N is flat and, by assumption, a D41-module, then 𝑔 splits, which implies that M is projective. Consequently, by [24, Lemma 10], we conclude that R is a left perfect ring. 5) ⇒ 1). By [4, Corollary 3.4], RN is a lifting module. So, by [1, Theorem 1.2.17], R is a right perfect ring. Definition 4.9. A ring R is called strongly left D41 if (RR) n satisfies the D41 condition for every positive integer n ; that is, every finitely generated free left R-module is a D41-module. A left R- module M is called strongly left D41 if Mn is a D41-module for every positive integer n. Proposition 4.10. The following conditions are equivalent for a ring R: 1) R is Artinian semisimple ; 2) Every (finitely generated) left R-module is a D41-module ; 3) Every (finitely generated) left R-module is a strongly D41-module ; 4) Every 2-generated left R- module is a strongly D41-module ; 5) The class of all left D41-modules is closed under (finite) direct sums ; 6) The class of all strongly left D41-modules is closed under (finite) direct sums. Proof: The implications 1) ⇔ 2) ⇔ 5) follow from [23, Theorem 9]. 6) ⇒ 4, 3) ⇒ 4) and 1) ⇒ 3) ⇒ 2) are obvious. 6) ⇒ 1) Let M be a simple R-module. This implies that M is a strongly D41-module. Given the hypothesis, R ⊕ M is also strongly D41. We note that M is isomorphic to R/J for some maximal left ideal J of R. Thus, there exists an epi-morphism 𝑅 → 𝑀 → 0. By Proposition 3.5, we conclude that M is a D41-module. Therefore, R is semisimple. 4) ⇒ 1) Since every simple R-module M is a strongly D41-module and R ⊕ M is 2-generated, the claim can be proved using an argument similar to that in the proof of 6) ⇒ 1). Proposition 4.11. The following conditions are equivalent for a ring R: 1) R is a semisimple Artinian ring. 2) Every left R-module has a D41-cover. 3) Every 2-generated left R-module has a D41-cover. 4) Every left R-module has a D41-envelope. 5) Every 2-generated left R-module has a D41-envelope. Proof: The implications 1) ⇒ 2) ⇒ 3) and 1) ⇒ 4) ⇒ 5) are straightforward. 3) ⇒ 1). Let 𝑆 be a simple left R-module, and denote 𝜓 ∶ 𝑅 𝑅 → 𝑆 as an epimorphism. By condition 3), M =RR ⊕ S has a D41-cover, which we can denote as 𝜆 ∶ 𝐶 → 𝑀, where C is a D41-module. Let Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 690 https://internationalpubls.com 𝛿1 ∶ 𝑆 → 𝑀 and 𝛿2 :RR → M be the inclusion maps for i = 1, 2. Notably, both 𝑆 and RR are D41- modules, and there exist homomorphisms 𝛾1 ∶ 𝑆 → 𝐶 and 𝛾2 :RR → C such that 𝜆 ∘ 𝛾𝑖 = 𝛿𝑖. Clearly, we have idM = δ1 ⊕ δ2 = λ ◦ (γ1 ⊕ γ2). This shows that M is isomorphic to a direct summand of C , implying that M is a D41-module. Consequently, we conclude that 𝑘𝑒𝑟(𝜓) is a direct summand of RR by [24, Proposition 4]. Therefore, S is a projective module, leading us to conclude that R is semisimple. 5) ⇒ 1). Let S be a simple right R-module, and let ψ :RR → S be an epimorphism. By condition 5), M =R R ⊕ S has a D41-envelope, denoted by α : M → K , where K is a D41-module. Since both S and R are D41-modules, there exist homo-morphisms 𝑓1 ∶ 𝐾 → 𝑆 𝑎𝑛𝑑 𝑓2 ∶ 𝐾 → 𝑅 such that 𝑓𝑖𝛼 = 𝜋𝑖, where 𝜋1 ∶ 𝑀 → 𝑆 and 𝜋2 ∶ 𝑀 → 𝑅 are the projection maps. Additionally, there exists a homomorphism 𝜙 ∶ 𝐾 → 𝑀 such that 𝜋𝑖𝜙 = 𝑓𝑖 for i = 1, 2. This implies that ϕα = idM , making α a split monomorphism. Thus, S ⊕E (S) is isomorphic to a direct summand of K , indicating that S ⊕R is also a D41-module. From this, we conclude that 𝑘𝑒𝑟(𝜓) is a direct summand of RR. Therefore, S is a projective module, leading us to the conclusion that R is semisimple. Proposition 4.12. The following conditions are equivalent for a ring R: 1) R is hereditary (or semihereditary). 2) Every submodule (or finitely generated submodule) of a projective R-module is a D41-module. 3) Every submodule (or finitely generated submodule) of a projective R-module is a strongly D41- module. Proof: The implications 1) ⇒ 3) ⇒ 2) and 1) ⇒ 2) are evident. 2) ⇒ 1). Let K be a projective submodule of an R-module M . Consider a free R-module F and an epimorphism 𝑔 ∶ 𝐹 → 𝐾. Then F ⊕ K is a projective submodule of F ⊕ M , which implies that F ⊕ K is a D41-module. Consequently, K is a pure direct-projective module, leading to the conclusion that the epimorphism 𝑔 ∶ 𝐹 → 𝐾 → 0 splits. 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