Microsoft Word - 167-177 Β  Β  167 Β  Ibn Al-Haitham Jour.for Pure&Appl.Sci. IHJPAS https://doi.org/10.30526/32.1.1919 Vol. 32 (1) 2019 Strongly π“š-nonsingular Modules Tha'ar Younis Ghawi thar.younis@qu.edu.iq Department of Mathematics, College of Education, AL-Qadisiyah University AL-Qadisiyah, Iraq. Article history: Received 12 August 2018, Accepted 26 September 2018, Publish January 2019 Abstract A submodule N of a module M is said to be s-essential if it has nonzero intersection with any nonzero small submodule in M. In this article, we introduce and study a class of modules in which all its nonzero endomorphisms have non-s-essential kernels, named, strongly 𝒦-nonsigular. We investigate some properties of strongly 𝒦-nonsigular modules. Direct summand, direct sums and some connections of such modules are discussed. Keywords: Modules; S-essential submodules; nonsingular modules; Strongly 𝒦-nonsigular modules. 1. Introduction A proper submodule N of a module M is said to be small if for any submodule K of M with 𝑁 𝐾 𝑀 implies 𝐾 𝑀[1]. A nonzero module M is called Hollow if all its proper submodules are small [2]. The dual concept of small submodule is an essential submodule, where a nonzero submodule N of a module M is called essential if for any submodule K of M with 𝑁 ∩ 𝐾 0 implies 𝐾 0. A nonzero R-module M is said to be uniform if all its nonzero submodules are essential [3]. As mixing of concepts small and essential submodules, we introduced the following class of submodules. A submodule N of M is said to be s-essential if for any small K in M with 𝑁 ∩ 𝐾 0 implies 𝐾 0 [4]. It is clear essential submdules implies s-essential. Roman C.S. in [5], recall that an R-module M is called 𝒦-nonsigular if for any endomorphism πœ‘ of M which has essential kernel, πœ‘ 0. 𝒦-a nonsingular module is studied in detail by [6]. In this research, we introduced concept of strongly 𝒦-nonsigular modules which is stronger than 𝒦-nonsigular modules. An R-module M is said to be strongly 𝒦-nonsigular if for each endomorphism of M which has s-essential kernel, is zero. In section 2, we give some characterizations and properties of this concept. In section 3, we proved a strongly 𝒦-nonsigular module is inherited by direct summands. Also, we give a condition for finite direct sums of strongly 𝒦-nonsigular modules to be strongly 𝒦-nonsigular. Several connections between strongly 𝒦-nonsigular and other classes, also some examples are proved in section 4. Throughout this work, all rings are associative with identity and all modules are unitary right R-modules. For a right R-module M, the notations𝑁 βŠ† 𝑀, 𝑁 𝑀, 𝑁 β‰ͺ 𝑀, 𝑁 ⊴ 𝑀, 𝑁 ⊴ 𝑀 or 𝑁 ⨁ 𝑀 denotes that N is a subset, a submodule, a small submodule, an essential submodule, a s-essential submodule, or direct summand of M, Β  Β  168 Β  Ibn Al-Haitham Jour.for Pure&Appl.Sci. IHJPAS https://doi.org/10.30526/32.1.1919 Vol. 32 (1) 2019 respectively. Also, for 𝑁 𝑀, we denote the endomorphism ring of M by 𝐸𝑛𝑑 𝑀 , π‘Ÿ 𝑁 π‘Ÿ ∈ 𝑅| π‘π‘Ÿ 0 and 𝑁: 𝑀 π‘Ÿ ∈ 𝑅| π‘€π‘Ÿ βŠ† 𝑁 . Starting, we will state some properties of s-essential submodules in [4, Prop. 2.7] which needed in this work. Proposition 1: Let M be a module. Then; (1) Assume 𝑁, 𝐾, 𝐿 are submodules of M with 𝐾 𝑁. 𝑖 If 𝐾 ⊴ 𝑀, then 𝐾 ⊴ 𝑁 and 𝑁 ⊴ 𝑀. 𝑖𝑖 𝑁 ⊴ 𝑀 and 𝐿 ⊴ 𝑀 if and only if 𝑁 ∩ 𝐿 ⊴ 𝑀. (2) If πœ‘: 𝑀 β†’ 𝑀 is a homomorphism with 𝐾 ⊴ 𝑀, then πœ‘ 𝐾 ⊴ 𝑀. (3) If 𝐾 βŠ† 𝑀 βŠ† 𝑀, 𝐾 βŠ† 𝑀 βŠ† 𝑀 and 𝑀 𝑀 ⨁𝑀 . Then 𝐾 ⨁𝐾 ⊴ 𝑀 ⨁𝑀 if and only if 𝐾 ⊴ 𝑀 for 𝑖 1,2. 2. Strongly π“š-nonsigular Modules In this section, we introduce the class of strongly 𝒦-nonsigular modules as a stronger class of 𝒦-nonsigular modules. Several various properties are proved. Definition 2. An R-module M is said to be strongly 𝒦-nonsigular if for all πœ‘ ∈ 𝐸𝑛𝑑 𝑀 with π‘˜π‘’π‘Ÿπœ‘ is s-essential in M, implies πœ‘ 0. Also, a ring R is strongly 𝒦-nonsigular if it is a strongly 𝒦-nonsigular R-module. for 𝑁 𝑀, if π»π‘œπ‘š , 𝑀 0 then N is called quasi-invertible [7]. Firstly, we are now in a position to give a characterization the notion of strongly 𝒦-nonsigular modules. Theorem 3. A module M is strongly 𝒦-nonsigular if and only if all its s-essential submodules are quasi-invertible. Proof. Assume M is a strongly 𝒦-nonsigular R-module. Let 𝑁 ⊴ 𝑀 and N is not quasi- invertible, i.e. π»π‘œπ‘š , 𝑀 0, so there exists 0 πœ‘: β†’ 𝑀. Consider πœ“ πœ‘ ∘ πœ‹ ∈ 𝐸𝑛𝑑 𝑀 , where πœ‹ is a natural epimorphism map. It is clear that 𝑁 βŠ† π‘˜π‘’π‘Ÿπœ“, but 𝑁 ⊴ 𝑀, this implies π‘˜π‘’π‘Ÿπœ“ ⊴ 𝑀, and hence πœ“ 0, as M is strongly 𝒦-nonsigular, thus πœ‘ 0, a contradiction. Therefore 𝑁 ⊴ 𝑀 and N is quasi-invertible. Conversely, let 0 𝑓 ∈ 𝐸𝑛𝑑 𝑀 . If π‘˜π‘’π‘Ÿπ‘“ ⊴ 𝑀, so by hypothesis π‘˜π‘’π‘Ÿπ‘“ is quasi-invertible. But, we can define a homomorphism β„Ž: β†’ 𝑀 by β„Ž π‘š πΎπ‘’π‘Ÿπ‘“ 𝑓 π‘š for all π‘š ∈ 𝑀. So β„Ž 0 and hence π»π‘œπ‘š , 𝑀 0 which is a contradiction with π‘˜π‘’π‘Ÿπ‘“ is quasi-invertible. Therefore π‘˜π‘’π‘Ÿπ‘“ ⋬ 𝑀 and M is a strongly 𝒦-nonsigular R-module. ∎ Corollary 4. Let M be a strongly 𝒦-nonsigular module. If 𝑁 ⊴ 𝑀, then π‘Ÿ 𝑁 π‘Ÿ 𝑀 . Proof. Assume 𝑁 ⊴ 𝑀, then by previous Theorem, N is a quasi-invertible submodule, and so π‘Ÿ 𝑁 π‘Ÿ 𝑀 by [7, Prop. 1.1.4]. ∎ Β  Β  169 Β  Ibn Al-Haitham Jour.for Pure&Appl.Sci. IHJPAS https://doi.org/10.30526/32.1.1919 Vol. 32 (1) 2019 Proposition 5. Let M be an R-module, π‘…βˆ— 𝑅 𝐴⁄ and 𝐴 βŠ† π‘Ÿ 𝑀 . Then M is a strongly 𝒦- nonsingular R-module if and only if M is a strongly 𝒦-nonsigular π‘…βˆ—-module. Proof. Assume πœ‹: 𝑅 β†’ π‘…βˆ— is a natural epimorphism, so by [8, Ex. P.51] π»π‘œπ‘š , 𝑀 π»π‘œπ‘š βˆ— , 𝑀 for each submodule N of M. So, the result is follow. ∎ Proposition 6. Let M be a strongly 𝒦-nonsigular module with 𝑀 𝑋⁄ is a projective module for all 𝑋 ⊴ 𝑀. Then 𝑀 𝐴⁄ is a strongly 𝒦-nonsigular module, for all 𝐴 ⊴ 𝑀. Proof. For 𝐡 𝐴 ⊴ 𝑀 𝐴⁄⁄ , to prove that π»π‘œπ‘š ⁄ ⁄ , 0, that is; π»π‘œπ‘š , 0. If false, so there is a nonzero homomorphism πœ‘: β†’ . Note that 𝐡 ⊴ 𝑀 (in fact, 𝐴 βŠ† 𝐡 βŠ† 𝑀 with 𝐴 ⊴ 𝑀), so by hypothesis 𝑀 𝐡⁄ is projective, hence there is a homomorphism πœ“: β†’ 𝑀 such that πœ‘ πœ‹ ∘ πœ“. It is clear πœ“ 0, this implies π»π‘œπ‘š , 𝑀 0 with 𝐡 ⊴ 𝑀, is a contradiction with M is strongly 𝒦-nonsigular. Thus πœ‘ 0 and 𝑀 𝐴⁄ is a strongly 𝒦-nonsigular R-module. ∎ Definition 7. Let M be a module, define the 𝑠-𝒦-nonsigular submodule of M by 𝑍 𝒦 𝑀 βˆ‘ πΌπ‘šπœ‘βˆˆ , where 𝑆 𝐸𝑛𝑑 𝑀 and π‘˜π‘’π‘Ÿπœ‘ ⊴ 𝑀. Now, we will give another characterization for a strongly 𝒦-nonsigular module as follows. Proposition 8. Let M be a module. Then M is strongly 𝒦-nonsigular if and only if 𝑍 𝒦 𝑀 0. Proof. If M is a strongly 𝒦-nonsigular module, then for all πœ‘ ∈ 𝐸𝑛𝑑 𝑀 with π‘˜π‘’π‘Ÿπœ‘ ⊴ 𝑀, implies πΌπ‘šπœ‘ 0, and hence 𝑍 𝒦 𝑀 βˆ‘ πΌπ‘šπœ‘ 0∈ , where 𝑆 𝐸𝑛𝑑 𝑀 and π‘˜π‘’π‘Ÿπœ‘ ⊴ 𝑀. Conversely, assume 𝑍 𝒦 𝑀 0. Let πœ“ ∈ 𝐸𝑛𝑑 𝑀 such that π‘˜π‘’π‘Ÿπœ“ ⊴ 𝑀, then πΌπ‘šπœ“ βŠ† 𝑍 𝒦 𝑀 and so πœ“ 0. Hence M is a strongly 𝒦-nonsigular module. ∎ Let M be a module, recall that a submodule N is supplement of 𝐾 𝑀 if, N is a minimal in the set of submodules 𝐿 𝑀 with 𝐾 𝐿 𝑀 (Equivalently, N is supplement of 𝐾 𝑀 if and only if 𝐾 𝑁 𝑀 and 𝐾 ∩ 𝑁 β‰ͺ 𝑁) [9]. We say that a submodule N of a module M is a supplement if it is a supplement for some submodule L of M. The transitive property of s-essential submodules need not be hold, see [4, Ex. 2.8]. So, we will give a condition for which the transitive property is hold of s-essential submodules. Lemma 9. Let M be a module, and let N is a supplement submodule in M with 𝐾 βŠ† 𝑁 βŠ† 𝑀. If 𝐾 ⊴ 𝑁 and 𝑁 ⊴ 𝑀, then 𝐾 ⊴ 𝑀. Proof. Assume 𝐿 β‰ͺ 𝑀 with 𝐾 ∩ 𝐿 0. If 𝐿 βŠ† 𝑁, but N is a supplement in M, then by [10, Prop. 20.2] 𝐿 β‰ͺ 𝑁, and hence 𝐿 0, since 𝐾 ⊴ 𝑁. Now, if 𝐿 ⊈ 𝑁. We have 𝐿 ∩ 𝑁 βŠ† 𝑁 βŠ† 𝑀, but (𝐿 β‰ͺ 𝑀 implies 𝐿 ∩ 𝑁 β‰ͺ 𝑀), thus again by [10, Prop. 20.2] 𝐿 ∩ 𝑁 β‰ͺ 𝑁, since N is a supplement in M. But 𝐾 ∩ 𝐿 ∩ 𝑁 𝐾 ∩ 𝐿 0 and 𝐾 ⊴ 𝑁, this implies 𝐿 ∩ 𝑁 0, and hence 𝐿 0, as 𝑁 ⊴ 𝑀 . ∎ Now, we present the following Proposition. Β  Β  170 Β  Ibn Al-Haitham Jour.for Pure&Appl.Sci. IHJPAS https://doi.org/10.30526/32.1.1919 Vol. 32 (1) 2019 Proposition 10. Let M be a quasi-injective R-module, and let N is a s-essential and supplement submodule in M. If M is a strongly 𝒦-nonsigular R-module, then so is N. Proof. Let 0 𝑓: 𝑁 β†’ 𝑁 be a homomrphism. Since M is a quasi-injective module, there exists 0 πœ‘ ∈ 𝐸𝑛𝑑 𝑀 such that 𝑖 ∘ 𝑓 πœ‘ ∘ 𝑖, where 𝑖: 𝑁 β†’ 𝑀 is an inclusion map. As M is strongly 𝒦-nonsigular, we get π‘˜π‘’π‘Ÿπœ‘ ⋬ 𝑀. Clearly, π‘˜π‘’π‘Ÿπ‘“ βŠ† π‘˜π‘’π‘Ÿπœ‘ then π‘˜π‘’π‘Ÿπ‘“ ⋬ 𝑀. If π‘˜π‘’π‘Ÿπ‘“ ⊴ 𝑁, and since 𝑁(supplement) ⊴ 𝑀, so by previous Lemma, π‘˜π‘’π‘Ÿπ‘“ ⊴ 𝑀, is a contradiction. Therefore π‘˜π‘’π‘Ÿπ‘“ ⋬ 𝑁, and N is a strongly 𝒦-nonsigular module. ∎ A quasi-injective module 𝑀 is called quasi-injective hull of a module M if, there exists a monomorphism πœ‘: 𝑀 β†’ 𝑀 with πΌπ‘šπœ‘ ⊴ 𝑀 [11]. Corollary 11. Let 𝑀 be a strongly 𝒦-nonsigular module. If M is a supplement in 𝑀, then M is strongly 𝒦-nonsigular. Next, we will study the behavior of s-essential submodule and strongly 𝒦-nonsigular module under localization. Firstly, we have the following Lemma. Lemma 12. Let M be a module, 𝑁 𝐾 𝑀 and let S is a multiplicative closed subset of R, provided 𝑆 𝐿 𝑆 𝐿 iff 𝐿 𝐿 for all 𝐿 , 𝐿 𝑀. Then the following hold. 𝑖 𝑁 β‰ͺ 𝐾 in M as R-module if and only if 𝑆 𝑁 β‰ͺ 𝑆 𝐾 in 𝑆 𝑀 as 𝑆 𝑅-module. 𝑖𝑖 𝑁 ⊴ 𝐾 in M as R-module if and only if 𝑆 𝑁 ⊴ 𝑆 𝐾 in 𝑆 𝑀 as 𝑆 𝑅-module. Proof. 𝑖 Assume 𝑁 β‰ͺ 𝐾 𝑀. Let 𝑆 𝐿 𝑆 𝐾 with 𝑆 𝑁 𝑆 𝐿 𝑆 𝐾, where 𝐿 𝐾. But we have 𝑆 𝑁 𝑆 𝐿 𝑆 𝑁 𝐿 , so 𝑆 𝑁 𝐿 𝑆 𝐾, and hence 𝑁 𝐿 𝐾 by hypothesis, thus 𝐿 𝐾, as 𝑁 β‰ͺ 𝐾. Therefore 𝑆 𝐿 𝑆 𝐾, and so 𝑆 𝑁 β‰ͺ 𝑆 𝐾 in 𝑆 𝑀. Conversely, if 𝑁 𝐿 𝐾 where 𝐿 𝐾. Then 𝑆 𝑁 𝑆 𝐿 𝑆 𝑁 𝐿 𝑆 𝐾, and hence 𝑆 𝐿 𝑆 𝐾, as 𝑆 𝑁 β‰ͺ 𝑆 𝐾. By hypothesis, 𝐿 𝐾, and so 𝑁 β‰ͺ 𝐾 in M. 𝑖𝑖 If 𝑁 ⊴ 𝐾 𝑀. Let 𝑆 𝐿 β‰ͺ 𝑆 𝐾 such that 𝑆 𝑁 ∩ 𝑆 𝐿 𝑆 0, where 𝐿 𝐾. By 𝑖 , 𝐿 β‰ͺ 𝐾. But, we have 𝑆 𝑁 ∩ 𝐿 𝑆 𝑁 ∩ 𝑆 𝐿 𝑆 0, 𝑁 ∩ 𝐿 0 by hypothesis. As 𝑁 ⊴ 𝐾 and 𝐿 β‰ͺ 𝐾 implies 𝐿 0, thus 𝑆 𝐿 𝑆 0. Conversely, suppose 𝑁 ∩ 𝐿 0 where 𝐿 β‰ͺ 𝐾, implies 𝑆 𝐿 β‰ͺ 𝑆 𝐾, by 𝑖 . So 𝑆 𝑁 ∩ 𝑆 𝐿 𝑆 𝑁 ∩ 𝐿 𝑆 0, thus 𝑆 𝐿 𝑆 0, as 𝑆 𝑁 ⊴ 𝑆 𝐾. By hypothesis, 𝐿 0. ∎ However, we get the following result. Proposition 13. Let M be an R-module, and let S is a multiplicative closed subset of R such that 𝑆 𝐿 𝑆 𝐾 iff 𝐿 𝐾 for all 𝐿, 𝐾 𝑀. Then M is a strongly 𝒦-nonsigular R-module, whenever 𝑆 𝑀 is a strongly 𝒦-nonsigular 𝑆 𝑅-module. Proof. Assume 0 𝑔 ∈ 𝐸𝑛𝑑 𝑀 . We can define an 𝑆 𝑅-homomorphism 𝑆 𝑔: 𝑆 𝑀 β†’ 𝑆 𝑀 such that 𝑆 𝑔 for each π‘š ∈ 𝑀, 𝑠 ∈ 𝑆. It is clear 𝑆 𝑔 0, so π‘˜π‘’π‘Ÿ 𝑆 𝑔 ⋬ 𝑆 𝑀, as 𝑆 𝑀 is strongly 𝒦-nonsigular. Also, it is easy to see that π‘˜π‘’π‘Ÿ 𝑆 𝑔 𝑆 π‘˜π‘’π‘Ÿπ‘” , this implies that 𝑆 π‘˜π‘’π‘Ÿπ‘” ⋬ 𝑆 𝑀, and hence by Lemma 12 𝑖𝑖 , π‘˜π‘’π‘Ÿπ‘” ⋬ 𝑀. ∎ Β  Β  171 Β  Ibn Al-Haitham Jour.for Pure&Appl.Sci. IHJPAS https://doi.org/10.30526/32.1.1919 Vol. 32 (1) 2019 Proposition 14. Let M be an R-module, and let P is a maximal ideal of R. If 𝑀 is a strongly 𝒦- nonsigular 𝑅 -module, then M is a strongly 𝒦-nonsigular R-module. Recall that an R-module M is called multiplication if for each submodule N of M, 𝑁 𝑀𝐼 for some ideal I of R (Equivalently, M a multiplication if and only if 𝑁 𝑀. 𝑁: 𝑀 ) [12]. If π‘Ÿ 𝑀 0, then M is called a faithful R-module. An R-module M is said to be scalar if for any πœ‘ ∈ 𝐸𝑛𝑑 𝑀 , πœ‘ π‘š π‘šπ‘Ÿ for some π‘Ÿ ∈ 𝑅, and for all π‘š ∈ 𝑀 [13]. Now, we will studied the strongly 𝒦-nonsigular property for rings and modules. But, in a position we need the following Lemma. Lemma 15. The following holds, for faithful multiplication R-module M. 𝑖 𝑁 β‰ͺ 𝑀 if and only if 𝐼 β‰ͺ 𝑅, where 𝑁 𝑀𝐼. 𝑖𝑖 𝑁 ⊴ 𝑀 if and only if 𝐼 ⊴ 𝑅, where 𝑁 𝑀𝐼. Proof. 𝑖 Assume that 𝑁 β‰ͺ 𝑀. Let J be any ideal of R with 𝐼 𝐽 𝑅, so 𝑀 𝐼 𝐽 𝑀𝑅, that is; 𝑁 𝑀𝐽 𝑀, but 𝑁 β‰ͺ 𝑀 implies 𝑀𝐽 𝑀, and so 𝐽 𝑅, since M is a faithful multiplication R-module. Thus 𝐼 β‰ͺ 𝑅. Conversely, let 𝐾 𝑀 with 𝑁 𝐾 𝑀. As M is multiplication, 𝐾 𝑀𝐽 for some 𝐽 𝑅. Hence 𝑀 𝐼 𝐽 𝑁 𝐾 𝑀 𝑀𝑅, but M is a faithful multiplication R- module, so 𝐼 𝐽 𝑅, thus 𝐽 𝑅 (since 𝐼 β‰ͺ 𝑅). Therefore, 𝐾 𝑀𝐽 𝑀𝑅 𝑀, and hence 𝑁 β‰ͺ 𝑀. 𝑖𝑖 Let 𝑁 ⊴ 𝑀. Suppose that 𝐽 β‰ͺ 𝑅 with 𝐼 ∩ 𝐽 0, then 𝑁 ∩ 𝑀𝐽 𝑀𝐼 ∩ 𝑀𝐽 𝑀 𝐼 ∩ 𝐽 0, but by 𝑖 , 𝑀𝐽 β‰ͺ 𝑀, hence 𝑀𝐽 0, implies 𝐽 0 (since M is faithful). Thus 𝐼 ⊴ 𝑅. Conversely, let 𝐾 β‰ͺ 𝑀 such that 𝑁 ∩ 𝐾 0. Since M is multiplication, then there is a small ideal J of R with 𝐾 𝑀𝐽, by 𝑖 . Hence 𝑀 𝐼 ∩ 𝐽 𝑀𝐼 ∩ 𝑀𝐽 𝑁 ∩ 𝐾 0, so by faithfulty for M, we get 𝐼 ∩ 𝐽 0, then 𝐽 0, as 𝐽 β‰ͺ 𝑅 and 𝐼 ⊴ 𝑅. Thus 𝐾 𝑀𝐽 0, and so 𝑁 ⊴ 𝑀. ∎ Proposition 16. Let M be a faithful multiplication R-module. If M is a strongly 𝒦-nonsigular R- module, then R is strongly 𝒦-nonsigular. The converse hold, whenever M is finitely generated. Proof. Assume that M is a strongly 𝒦-nonsigular R-module. Let 0 πœ‘ ∈ 𝐸𝑛𝑑 𝑅 . For π‘Ÿ ∈ 𝑅, we know πœ‘ π‘Ž π‘Ž. πœ‘ 1 . We can define πœ“: 𝑀 β†’ 𝑀 by πœ“ π‘š π‘š. πœ‘ 1 for all π‘š ∈ 𝑀. It is easy to see πœ“ is well-defined and homomorphism. If πœ“ 0, then 𝑀. πœ‘ 1 0, hence πœ‘ 1 ∈ π‘Ÿ 𝑀 0, so πœ‘ 0 which is a contradiction. Hence 0 πœ“ ∈ 𝐸𝑛𝑑 𝑀 , and so π‘˜π‘’π‘Ÿπœ“ ⋬ 𝑀, as M is strongly 𝒦-nonsigular. Since M is a multiplication R-module, π‘˜π‘’π‘Ÿπœ“ 𝑀. π‘˜π‘’π‘Ÿπœ“: 𝑀 . But, we have π‘˜π‘’π‘Ÿπœ“: 𝑀 π‘˜π‘’π‘Ÿπœ‘, to see this: if π‘Ÿ ∈ π‘˜π‘’π‘Ÿπœ“: 𝑀 , π‘€π‘Ÿ βŠ† π‘˜π‘’π‘Ÿπœ“, so πœ“ π‘€π‘Ÿ π‘€π‘Ÿ. πœ‘ 1 𝑀. πœ‘ π‘Ÿ 0, hence πœ‘ π‘Ÿ ∈ π‘Ÿ 𝑀 0, thus π‘Ÿ ∈ π‘˜π‘’π‘Ÿπœ‘. Now, if π‘₯ ∈ π‘˜π‘’π‘Ÿπœ‘, πœ‘ π‘₯ π‘₯. πœ‘ 1 0 hence 𝑀π‘₯. πœ‘ 1 0, so πœ“ 𝑀π‘₯ 0 implies 𝑀π‘₯ βŠ† π‘˜π‘’π‘Ÿπœ“, thus π‘₯ ∈ π‘˜π‘’π‘Ÿπœ“: 𝑀 . Since π‘˜π‘’π‘Ÿπœ“ ⋬ 𝑀, so 𝑀. π‘˜π‘’π‘Ÿπœ“: 𝑀 ⋬ 𝑀, so by Lemma 15 𝑖𝑖 , π‘˜π‘’π‘Ÿπœ“: 𝑀 ⋬ 𝑅, which hence π‘˜π‘’π‘Ÿπœ‘ ⋬ 𝑅, therefore R is strongly 𝒦-nonsigular. Conversely, let 0 𝑔 ∈ 𝐸𝑛𝑑 𝑀 . If M is finitely generated multiplication R-module, then M is a scalar R-module, by [14, Th. 2.3]. Hence 𝑔 π‘š π‘šπ‘Ÿ for some π‘Ÿ ∈ 𝑅, and for all π‘š ∈ 𝑀. It follows that β„Ž ∈ 𝐸𝑛𝑑 𝑅 defined by β„Ž π‘₯ π‘₯π‘Ÿ for all π‘₯ ∈ 𝑅. Note β„Ž 1 1. π‘Ÿ π‘Ÿ 0 (in fact, if π‘Ÿ 0 implies 𝑔 0), and hence 0 β„Ž ∈ 𝐸𝑛𝑑 𝑅 , but R is strongly 𝒦-nonsigular, then π‘˜π‘’π‘Ÿβ„Ž ⋬ 𝑅. On the other hand, we have Β  Β  172 Β  Ibn Al-Haitham Jour.for Pure&Appl.Sci. IHJPAS https://doi.org/10.30526/32.1.1919 Vol. 32 (1) 2019 π‘˜π‘’π‘Ÿβ„Ž π‘˜π‘’π‘Ÿπ‘”: 𝑀 which implies π‘˜π‘’π‘Ÿπ‘”: 𝑀 ⋬ 𝑅, and hence 𝑀. π‘˜π‘’π‘Ÿπ‘”: 𝑀 ⋬ 𝑀, by Lemma 15 𝑖𝑖 , thus π‘˜π‘’π‘Ÿπ‘” ⋬ 𝑀, and M is a strongly 𝒦-nonsigular R-module. ∎ Next, proved that the property of strongly 𝒦-nonsigular of modules is inherited by isomorphism. Proposition 17. For two modules 𝑀 and 𝑀 , if 𝑀 β‰… 𝑀 then 𝑀 is a strongly 𝒦-nonsigular module, whenever 𝑀 is strongly 𝒦-nonsigular. Proof. Since 𝑀 β‰… 𝑀 , there exists an isomorphism 𝑓: 𝑀 ⟢ 𝑀 . Assume 𝑀 is a strongly 𝒦- nonsigular module. Let 𝑔 ∈ 𝐸𝑛𝑑 𝑀 such that π‘˜π‘’π‘Ÿπ‘” ⊴ 𝑀 . Consider πœ“ 𝑓 ∘ 𝑔 ∘ 𝑓 ∈ 𝐸𝑛𝑑 𝑀 , where 𝑓 : 𝑀 ⟢ 𝑀 isomorphism. Now, we have π‘˜π‘’π‘Ÿπœ“ 𝑓 π‘˜π‘’π‘Ÿπ‘” , to see this: π‘˜π‘’π‘Ÿπœ“ π‘₯ ∈ 𝑀 | 𝑓 ∘ 𝑔 ∘ 𝑓 π‘₯ 0 π‘₯ ∈ 𝑀 | 𝑔 ∘ 𝑓 π‘₯ ∈ π‘˜π‘’π‘Ÿπ‘“ 0 π‘₯ ∈ 𝑀 | 𝑓 π‘₯ ∈ π‘˜π‘’π‘Ÿπ‘” π‘₯ ∈ 𝑀 | π‘₯ ∈ 𝑓 π‘˜π‘’π‘Ÿπ‘” 𝑓 π‘˜π‘’π‘Ÿπ‘” . By Proposition 1.1(2), we get 𝑓 π‘˜π‘’π‘Ÿπ‘” ⊴ 𝑀 , (since π‘˜π‘’π‘Ÿπ‘” ⊴ 𝑀 ), this implies π‘˜π‘’π‘Ÿπœ“ ⊴ 𝑀 and hence πœ“ 0, as 𝑀 is strongly 𝒦-nonsigular. Thus, 0 𝑓 ∘ 𝑔 πΌπ‘šπ‘“ 𝑓 ∘ 𝑔 𝑀 , thus πΌπ‘šπ‘” βŠ† π‘˜π‘’π‘Ÿπ‘“ 0. Therefore 𝑔 0. ∎ Proposition 18. Let M be a faithful scalar R-module. Then R is strongly 𝒦-nonsigular if and only if 𝑆 𝐸𝑛𝑑 𝑀 is strongly 𝒦-nonsigular. Proof. Since M is a scalar R-module, then by [15, Lemma 3.6.2] 𝑆 𝐸𝑛𝑑 𝑀 β‰… 𝑅 π‘Ÿ 𝑀⁄ , but M is faithful, hence 𝑆 𝐸𝑛𝑑 𝑀 β‰… 𝑅. By Proposition 17, the result is follow. ∎ Proposition 19. Let M be a faithful multiplication R-module. If R is strongly 𝒦-nonsigular, then π‘Ÿ 𝑁 π‘Ÿ 𝑀 for all 𝑁 ⊴ 𝑀. Proof. As M is a faithful multiplication R-module, if 𝑁 ⊴ 𝑀, there is 𝐼 ⊴ 𝑅 with 𝑁 𝑀𝐼, by Lemma 15 𝑖𝑖 . For π‘Ÿ ∈ π‘Ÿ 𝑁 , π‘π‘Ÿ 0, then 𝑀𝐼. π‘Ÿ 0, hence πΌπ‘Ÿ βŠ† π‘Ÿ 𝑀 0, so π‘Ÿ ∈ π‘Ÿ 𝐼 implies π‘Ÿ 𝑁 π‘Ÿ 𝐼 . Since R is strongly 𝒦-nonsigular with 𝐼 ⊴ 𝑅, then I is a quasi-invertible ideal (by Theorem 2.2), so π‘Ÿ 𝐼 π‘Ÿ 𝑅 0 by [7, Prop. 1.1.4]. Hence π‘Ÿ 𝑁 0 π‘Ÿ 𝑀 . ∎ 3. Direct Summand and Direct Sums We start with following result. Proposition 20. Let M be a strongly 𝒦-nonsigular module, and 𝐴 𝑀. If 𝐴 ⊴ 𝐡 ⨁ 𝑀, then 𝐡 𝐡 for 𝑖 ∈ 1,2 . Proof. Consider 𝜌 : 𝑀 ⟢ 𝐡 is the canonical projection map, for 𝑖 1,2. We have 𝜌 𝐴 𝐴 𝜌 𝐴 . Since 1 𝜌 𝜌 ∈ 𝐸𝑛𝑑 𝑀 , so we have 1 𝜌 𝜌 𝐴 1 𝜌 𝜌 𝐴 1 𝜌 𝜌 𝐴 1 𝜌 𝜌 𝐴 0 (since 𝜌 is an idempotent), then 𝐴 βŠ† π‘˜π‘’π‘Ÿ 1 𝜌 𝜌 . Now, 𝐡 ⨁ 𝑀, so 𝑀 𝐡 ⨁𝐡 for some 𝐡 𝑀. Hence 1 𝜌 𝜌 𝐡 1 𝜌 𝜌 𝐡 1 𝜌 0 0, thus 𝐡 βŠ† π‘˜π‘’π‘Ÿ 1 𝜌 𝜌 . Therefore 𝐡 ⨁𝐴 βŠ† π‘˜π‘’π‘Ÿ 1 𝜌 𝜌 . On the other hand, 𝐡 ⊴ 𝐡 and 𝐴 ⊴ 𝐡 , then 𝐡 ⨁𝐴 ⊴ 𝐡 ⨁𝐡 𝑀 by Proposition 1 (3), and Β  Β  173 Β  Ibn Al-Haitham Jour.for Pure&Appl.Sci. IHJPAS https://doi.org/10.30526/32.1.1919 Vol. 32 (1) 2019 so π‘˜π‘’π‘Ÿ 1 𝜌 𝜌 ⊴ 𝑀 which implies 1 𝜌 𝜌 0, as M is strongly 𝒦-nonsigular. Hence 𝜌 𝜌 𝜌 , so 𝐡 𝜌 𝐡 𝜌 𝜌 𝐡 𝜌 𝜌 𝐡 𝜌 𝐡 βŠ† 𝐡 β‡’ 𝐡 βŠ† 𝐡 . Similarly, taking 1 𝜌 𝜌 ∈ 𝐸𝑛𝑑 𝑀 , and we get 𝐡 βŠ† 𝐡 . ∎ Based on our result, we prove that direct summands of a strongly 𝒦-nonsigular module inherit the property. Proposition 21. A direct summand of a strongly 𝒦-nonsigular module is strongly 𝒦-nonsigular. Proof. Let 𝑀 be a strongly 𝒦-nonsigular module, and 𝐴 ⨁ 𝑀, so 𝑀 𝐴⨁𝐡 for some 𝐡 𝑀. Assume that 𝑓 ∈ 𝐸𝑛𝑑 𝐴 such that π‘˜π‘’π‘Ÿπ‘“ ⊴ 𝐴. Consider β„Ž 𝑖 ∘ 𝑓 ∘ 𝜌 ∈ 𝐸𝑛𝑑 𝑀 , where 𝜌 is the canonical projection map onto 𝐴, and i is the inclusion map from 𝐴 to 𝑀. So, we have πΎπ‘’π‘Ÿβ„Ž πΎπ‘’π‘Ÿπ‘“β¨π΅, to see this: for π‘₯ ∈ π‘˜π‘’π‘Ÿβ„Ž, π‘₯ π‘Ž 𝑏 where π‘Ž ∈ 𝐴 and 𝑏 ∈ 𝐡 with β„Ž π‘₯ 0, so 𝑓 π‘Ž 𝑖 ∘ 𝑓 π‘Ž 𝑖 ∘ 𝑓 𝜌 π‘₯ β„Ž π‘₯ 0, then π‘Ž ∈ π‘˜π‘’π‘Ÿπ‘“, and hence π‘₯ π‘Ž 𝑏 ∈ π‘˜π‘’π‘Ÿπ‘“ 𝐡, that is; π‘˜π‘’π‘Ÿβ„Ž π‘˜π‘’π‘Ÿπ‘“ 𝐡. On the other hand, π‘˜π‘’π‘Ÿπ‘“ ∩ 𝐡 βŠ† 𝐴 ∩ 𝐡 0, which implies π‘˜π‘’π‘Ÿβ„Ž π‘˜π‘’π‘Ÿπ‘“β¨π΅. Since π‘˜π‘’π‘Ÿπ‘“ ⊴ 𝐴 and 𝐡 ⊴ 𝐡, then π‘˜π‘’π‘Ÿβ„Ž π‘˜π‘’π‘Ÿπ‘“β¨π΅ ⊴ 𝐴⨁𝐡 𝑀 by Proposition 1.1(3). Thus β„Ž 0, as M strongly 𝒦-nonsigular. Hence πΌπ‘šπ‘“ 𝑓 𝐴 𝑖 ∘ 𝑓 𝐴 𝑖 ∘ 𝑓 𝜌 𝑀 β„Ž 𝑀 0. Therfore 𝑓 0 and 𝐴 is strongly 𝒦-nonsigular. ∎ Definition 22. Let M and N be two R-modules. Then M is called strongly 𝒦-nonsigular relative to N if, every πœ‘ ∈ π»π‘œπ‘š 𝑀, 𝑁 such that π‘˜π‘’π‘Ÿπœ‘ ⊴ 𝑀, implies πœ‘ 0. Obviously, M is strongly 𝒦- nonsigular if and only if M is strongly 𝒦-nonsigular relative to M. Proposition 23. If M is a strongly 𝒦-nonsigular module. For 𝑁 𝑀, M is strongly 𝒦-nonsigular relative to N. Proof. If 𝑁 𝑀, clear that M is strongly 𝒦-nonsigular relative to N. Assume that 𝑁 𝑀, if πœ“ ∈ π»π‘œπ‘š 𝑀, 𝑁 with π‘˜π‘’π‘Ÿπœ“ ⊴ 𝑀. Consider β„Ž 𝑖 ∘ πœ“, where 𝑖 is the inclusion map from N to M. So β„Ž ∈ 𝐸𝑛𝑑 𝑀 such that π‘˜π‘’π‘Ÿβ„Ž π‘˜π‘’π‘Ÿπœ“ ⊴ 𝑀, then β„Ž 0, as M is strongly 𝒦-nonsigular, hence πΌπ‘šπœ“ πœ“ 𝑀 𝑖 πœ“ 𝑀 β„Ž 𝑀 0, thus πœ“ 0. ∎ Lemma 24. For a module M, if 𝑁 ⊴ 𝐾 𝑀 for 𝑖 ∈∧ 1,2, … , 𝑛 , then β‹‚ 𝑁 ⊴ β‹‚ 𝐾 . Proof. Consider the case when the index set ∧ 1,2 . Let 𝑋 β‰ͺ 𝐾 ∩ 𝐾 with 𝑁 ∩ 𝑁 ∩ 𝑋 0, then 𝑁 ∩ 𝑁 ∩ 𝑋 0. Since 𝑋 β‰ͺ 𝐾 ∩ 𝐾 βŠ† 𝐾 , then 𝑋 β‰ͺ 𝐾 and hence 𝑁 ∩ 𝑋 β‰ͺ 𝐾 implies 𝑁 ∩ 𝑋 0, as 𝑁 ⊴ 𝐾 . Also, 𝑋 β‰ͺ 𝐾 and 𝑁 ⊴ 𝐾 , hence 𝑋 0. Thus 𝑁 ∩ 𝑁 ⊴ 𝐾 ∩ 𝐾 .∎ Theorem 25. Let 𝑀 𝑀 ⨁𝑀 be an R-module. Then M is strongly 𝒦-nonsigular if and only if 𝑀 is strongly 𝒦-nonsigular relative to 𝑀 , for 𝑖, 𝑗 ∈ 1,2 . Proof. Assume 𝑀 𝑀 ⨁𝑀 a strongly 𝒦-nonsigular module. By Proposition 21, 𝑀 is strongly 𝒦-nonsigular, for 𝑖 ∈ 1,2 . Hence 𝑀 is strongly 𝒦-nonsigular relative to 𝑀 , for 𝑖 ∈ 1,2 . Now, let πœ‘ ∈ π»π‘œπ‘š 𝑀 , 𝑀 such that π‘˜π‘’π‘Ÿπœ‘ ⊴ 𝑀 . Consider πœ“ 𝑖 ∘ πœ‘ ∘ 𝜌 ∈ 𝐸𝑛𝑑 𝑀 , where 𝜌 is Β  Β  174 Β  Ibn Al-Haitham Jour.for Pure&Appl.Sci. IHJPAS https://doi.org/10.30526/32.1.1919 Vol. 32 (1) 2019 the canonical projection map onto 𝑀 , 𝑖: 𝑀 ⟢ 𝑀 is the inclusion map. Clearly, π‘˜π‘’π‘Ÿπœ“ π‘˜π‘’π‘Ÿπœ‘β¨π‘€ , so π‘˜π‘’π‘Ÿπœ“ π‘˜π‘’π‘Ÿπœ‘β¨π‘€ ⊴ 𝑀 ⨁𝑀 𝑀, hence πœ“ 0 (since M is strongly 𝒦- nonsigular). Thus, πœ‘ 0 and so 𝑀 is strongly 𝒦-nonsigular relative to 𝑀 . 𝑀 is strongly 𝒦- nonsigular relative to 𝑀 , similarly. Conversely, if 𝑓 ∈ 𝐸𝑛𝑑 𝑀 such that π‘˜π‘’π‘Ÿπ‘“ ⊴ 𝑀, so we have π‘˜π‘’π‘Ÿπ‘“ ∩ 𝑀 ⊴ 𝑀 , by Lemma 24. Consider 𝑓| : 𝑀 β†’ 𝑀 which defined by 𝑓| π‘₯ 𝑓 π‘₯ 0 for all π‘₯ ∈ 𝑀. We have π‘˜π‘’π‘Ÿ 𝑓| π‘˜π‘’π‘Ÿπ‘“ ∩ 𝑀 as follows: if π‘Ž ∈ π‘˜π‘’π‘Ÿπ‘“ ∩ 𝑀 then 0 𝑓 π‘Ž 𝑓 π‘Ž 0 𝑓| π‘Ž and π‘Ž ∈ 𝑀 , thus π‘Ž ∈ π‘˜π‘’π‘Ÿ 𝑓| . Now, if π‘₯ ∈ π‘˜π‘’π‘Ÿ 𝑓| then 0 𝑓| π‘₯ 𝑓 π‘₯ 0 𝑓 π‘₯ , so π‘₯ ∈ π‘˜π‘’π‘Ÿπ‘“ ∩ 𝑀 . Consider 𝑔 𝜌 ∘ 𝑓| , where 𝜌 is the canonical projection map onto 𝑀 , for 𝑖 ∈ 1,2 . To prove that π‘˜π‘’π‘Ÿ 𝑓| β‹‚ π‘˜π‘’π‘Ÿπ‘” . If π‘₯ ∈ π‘˜π‘’π‘Ÿ 𝑓| , 0 𝑓| π‘₯ , so 𝑔 π‘₯ 𝜌 ∘ 𝑓| π‘₯ 𝜌 𝑓| π‘₯ 𝜌 0 0, this implies π‘₯ ∈ β‹‚ π‘˜π‘’π‘Ÿπ‘” . Now, if π‘₯ ∈ β‹‚ π‘˜π‘’π‘Ÿπ‘” , so 𝑔 π‘₯ 0 β‡’ 𝜌 𝑓| π‘₯ 0 β‡’ 𝑓| π‘₯ ∈ β‹‚ π‘˜π‘’π‘ŸπœŒ 𝑀 ∩ 𝑀 0 β‡’ π‘₯ ∈ π‘˜π‘’π‘Ÿ 𝑓| for 𝑖 ∈ 1,2 . So β‹‚ π‘˜π‘’π‘Ÿπ‘” π‘˜π‘’π‘Ÿ 𝑓| π‘˜π‘’π‘Ÿπ‘“ ∩ 𝑀 ⊴ 𝑀 , hence by Proposition 1, π‘˜π‘’π‘Ÿπ‘” ⊴ 𝑀 and π‘˜π‘’π‘Ÿπ‘” ⊴ 𝑀 . By hypothesis, 𝑔 0 β‡’ 𝜌 πΌπ‘š 𝑓| 0 β‡’ πΌπ‘šπ‘“| βŠ† β‹‚ π‘˜π‘’π‘ŸπœŒ 0 for 𝑖 ∈ 1,2 , implies 𝑓| 0. Similarly, we obtain β„Ž 𝜌 ∘ 𝑓| 0 for 𝑖 ∈ 1,2 , and hence 𝑓| 0. So 𝑓| 0 for 𝑖 ∈ 1,2 . Therefore 𝑓 0, and 𝑀 𝑀 ⨁𝑀 is strongly 𝒦-nonsigular. ∎ Corollary 26. If 𝑀 βŠ• 𝑀 . Then M is a strongly 𝒦-nonsigular module if and only if 𝑀 is strongly 𝒦-nonsigular relative to 𝑀 , for 𝑖, 𝑗 ∈ 1,2, … , 𝑛 . Proposition 27. Let 𝑀 𝑀 𝑀 be an R-module, where 𝑀 , 𝑀 𝑀. If ∩ is a strongly 𝒦-nonsigular R-module, then both of and is strongly 𝒦-nonsigular. Proof. We have ∩ ∩ ∩ ∩ , also ∩ ∩ ∩ ∩ ∩ 0 ∩ , thus ∩ ∩ ⨁ ∩ . As ∩ is strongly 𝒦-nonsigular, so by Proposition 3.2, ∩ is strongly 𝒦-nonsigular for 𝑖 1,2. But, we have ∩ β‰… and ∩ β‰… , so by Proposition 16, and are strongly 𝒦-nonsigular. ∎ 4. Connections to other Topics In this section, we can prove some relations between strongly 𝒦-nonsigular modules and other classes of modules, such examples, semisimple, Rickart, quasi-Dedekind and prime modules. Example 28. Every module has no nonzero small submodule, all its submodules are s-essential, and hence does not strongly 𝒦-nonsigular. Notice, every submodule in 𝑍 is s-essential, because the zero is the only small submodule of 𝑍 , hence 𝑍 is not strongly 𝒦-nonsigular. In particular, every simple (semisimple) module is not strongly 𝒦-nonsigular. But, we know every semisimple module is 𝒦-nonsigular. Β  Β  175 Β  Ibn Al-Haitham Jour.for Pure&Appl.Sci. IHJPAS https://doi.org/10.30526/32.1.1919 Vol. 32 (1) 2019 Remark 29. It is clear that every strongly 𝒦-nonsigular module is 𝒦-nonsigular, but the converse need not be true, in general, a semisimple module is 𝒦-nonsigular but not strongly 𝒦-nonsigular. Lemma 30. Let M be a Hollow (not simple) module, and 𝐴 𝑀. Then 𝐴 is essential if and only if 𝐴 is s-essential. Proof. β‡’ Clear. ⇐ Assume 0 𝐴 ⊴ 𝑀 such that 𝐴 ∩ 𝐡 0, where 𝐡 𝑀. If 𝐡 𝑀, then 𝐴 0, a contradiction. Thus B is a proper in M, hence 𝐡 β‰ͺ 𝑀 (since M is Hollow), and so 𝐡 0, as 𝐴 ⊴ 𝑀. Therfore 𝐴 ⊴ 𝑀. ∎ However, we consider the following Proposition by Lemma 30. Proposition 31. Let M be a Hollow (not simple) module. Then M is strongly 𝒦-nonsigular if and only if M is 𝒦-nonsigular. An R-module M is said to be Rickart if π‘Ÿ πœ‘ πΎπ‘’π‘Ÿπœ‘ is a direct summand of M for each πœ‘ ∈ 𝐸𝑛𝑑 𝑀 [16]. Recall that an R-module M is quasi-Dedekind if, for any 0 πœ‘ ∈ 𝐸𝑛𝑑 𝑀 , is a monomorphism (i.e. π‘˜π‘’π‘Ÿπœ‘ 0) [7]. Obviously, Rickart, quasi-Dedekind modules are 𝒦-nonsigular. Note that the Z-module 𝑍 is semisimple, so it is Rickart, but not strongly 𝒦-nonsigular. Also we know 𝑍 is quasi-Dedekind, but it is not strongly 𝒦-nonsigular. However, we have the following Corollary which follows by Proposition 4.4. Corollary 32. For a Hollow (not simple) module M. If M is Rickart (or quasi-Dedekind), then M is strongly 𝒦-nonsigular. Lemma 33. Let M be an R-module. If 𝑆 𝐸𝑛𝑑 𝑀 is a regular ring, then M is Rickart. Proof. Assume πœ‘ ∈ 𝑆 𝐸𝑛𝑑 𝑀 . Since 𝑆 is a regular ring, so πœ‘ a regular element, thus π‘˜π‘’π‘Ÿπœ‘ ⨁ 𝑀, by [17, Cor. 3.2]. Hence M is a Rickart module. ∎ Corollary 34. If M is a Hollow (not simple) R-module with 𝑆 𝐸𝑛𝑑 𝑀 is a regular ring, then M is strongly 𝒦-nonsigular. Proof. It follows directly by Lemma 33 and Corollary 34. ∎ Lemma 35. If M is a uniform module has nonzero small submodule, then s-essential submodule implies essential. Proof. Assume 𝑋 𝑀. Put 𝑋 0. Let N be a nonzero small submodule of M, then 𝑋 ∩ 𝑁 0 which implies 𝑋 ⋬ 𝑀. Hence the result is obtained. ∎ Note that Z-module Z is uniform, the zero submodule of 𝑍 is s-essential but not essential (in fact, 0 is the only small submodule of 𝑍 ). However, we have the following. Β  Β  176 Β  Ibn Al-Haitham Jour.for Pure&Appl.Sci. IHJPAS https://doi.org/10.30526/32.1.1919 Vol. 32 (1) 2019 Proposition 36. Let M be a uniform module has nonzero small submodule. Then M is strongly 𝒦- nonsigular if and only if M is 𝒦-nonsigular. Proof. It follows by Lemma 35. ∎ Recall [18], a module M is called prime if for all nonzero submodule N of M, π‘Ÿ 𝑁 π‘Ÿ 𝑀 . Mijbass in [7, Th. 2.3.14], presented the following Theorem. Theorem 37. A module 𝑀 is uniform quasi-Dedekind if and only if it is uniform prime. Proposition 38. Let 𝑀 be a uniform R-module has nonzero small submodule. Then the following asseretions are equivalent. 𝑖 𝑀 is Rickart. 𝑖𝑖 𝑀 is 𝒦-nonsigular. 𝑖𝑖𝑖 𝑀 is strongly 𝒦-nonsigular. 𝑖𝑣 𝑀 is quasi-Dedekind. 𝑣 𝑀 is prime. 𝑣𝑖 For 𝑁 ⊴ 𝑀, π‘Ÿ 𝑁 π‘Ÿ 𝑀 . Proof. 𝑖 β‡’ 𝑖𝑣 Since 𝑀 is a uniform R-module, then 𝑀 is indecomposable. Let πœ‘ ∈ 𝐸𝑛𝑑 𝑀 with πœ‘ 0, then π‘˜π‘’π‘Ÿπœ‘ ⨁ 𝑀, as 𝑀 is Rickart. So, either π‘˜π‘’π‘Ÿπœ‘ 𝑀 or π‘˜π‘’π‘Ÿπœ‘ 0. If π‘˜π‘’π‘Ÿπœ‘ 𝑀 then πœ‘ 0, a contradiction. Hence π‘˜π‘’π‘Ÿπœ‘ 0, implies 𝑀 is quasi-Dedekind. 𝑖𝑣 β‡’ 𝑖 Let πœ‘ ∈ 𝐸𝑛𝑑 𝑀 . If πœ‘ 0, then π‘˜π‘’π‘Ÿπœ‘ 𝑀 ⨁ 𝑀. Assume that πœ‘ 0, but 𝑀 is a quasi-Dedekind module, so π‘˜π‘’π‘Ÿπœ‘ 0 ⨁ 𝑀. Thus 𝑀 is Rickart. 𝑖𝑖 ⇔ 𝑖𝑖𝑖 It follows by Proposition 36. 𝑖𝑖 ⇔ 𝑖𝑣 Since 𝑀 is a uniform module, the result is follow. 𝑖𝑣 ⇔ 𝑣 It follows by Theorem 37. 𝑣 ⇔ 𝑣𝑖 Since 𝑀 is uniform has nonzero small submodule, then all its nonzero submodules are s-essential, so the result is obtained. ∎ 5. Conclusion The most important results of the article are: (1) Let M be a faithful multiplication R-module. If M is a strongly 𝒦-nonsigular R-module, then R is strongly 𝒦-nonsigular. The converse holds, whenever M is finitely generated. 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