Microsoft Word - 81-94   81  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (2) 2020       On Semisecond Submodules Zainab Saadi      Ghaleb Ahmed   Abstract Let 𝑀 be a right module over a ring 𝑅 with identity. The semisecond submodules are studied in this paper. A nonzero submodule 𝑁 of 𝑀 is called semisecond if 𝑁𝑎 𝑁𝑎 for each 𝑎 ∈ 𝑅. More information and characterizations about this concept is provided in our work. Keywords: semisecond submodules, weak semisecond submodules, 𝑆-semisecond submodules, regular modules. 1. Introduction 𝑅 is indicated a ring with identity and 𝑀 is viewed as a non-zero 𝑆- 𝑅-bimodule where 𝑆 𝐸𝑛𝑑 𝑀 the endomorphism ring of 𝑀. We use the notation ʻʻ ⊆ ʼʼ to denote inclusion. A non-zero submodule 𝑁 of 𝑀 is said to be a second submodule if for any 𝑎 ∈ 𝑅, the endomorphism 𝑓 : 𝑁 → 𝑁 defined by 𝑓 𝑛 𝑛𝑎 for each 𝑛 ∈ 𝑁, is either surjective or zero (that is 𝐼𝑚𝑓 𝑁𝑎 𝑁 or 𝐼𝑚𝑓 𝑁𝑎 0) [1]. Equivalently 0 𝑁 is a second submodule of 𝑀 if 𝑁𝐼 𝑁 or 𝑁𝐼 0 for every ideal 𝐼 of 𝑅 [1]. In that situation, 𝑎𝑛𝑛 𝑁 is a prime ideal of 𝑅[1]. A non-zero module 𝑀 is second (or coprime) if 𝑀 is a second submodule of itself [1]. As a new type of second submodules, the concept of weakly second submodules is presented in [2]. A non-zero submodule 𝑁 of 𝑀 is weakly second submodule whenever 𝑁𝑎𝑏 ⊆ 𝐾 where 𝑎, 𝑏 ∈ 𝑅 and 𝐾 a submodule of 𝑀 implies either 𝑁𝑎 ⊆ 𝐾 or 𝑁𝑏 ⊆ 𝐾 [2]. Equivalently, a non-zero submodule 𝑁 of 𝑀 is called weakly second if 𝑁𝑎𝑏 𝑁𝑎 or 𝑁𝑎𝑏 𝑁𝑏 for every 𝑎, 𝑏 ∈ 𝑅 [2]. More characterizations of the weakly second concept are provided in [3]. In fact this idea as a dual notion of the concept weakly prime (sometimes is called classical prime) submodules. A proper submodule 𝑁 of 𝑀 is wekly prime whenever 𝐾𝑎𝑏 ⊆ 𝑁 where 𝑎, 𝑏 ∈ 𝑅 and 𝐾 a submodule of 𝑀 implies either 𝐾𝑎 ⊆ 𝑁 or 𝐾𝑏 ⊆ 𝑁 [4]. In [5]. We define the idea of weakly secondary as a generalization of weakly second concept and the same time, it is a new class of secondary submodules and a dual notion of classical primary submodules respectively. A nonzero submodule 𝑁 of 𝑀 is weakly secondary submodule if Ibn Al Haitham Journal for Pure and Applied Science Journal homepage: http://jih.uobaghdad.edu.iq/index.php/j/index Doi: 10.30526/33.2.2429 zoozaih89@gmail.com ghaleb.a.h@ihcoedu.uobaghdad.edu.iq Article history: Received 2 June 2019, Accepted 15 July 2019, published April 2020. Department of Mathematics, College of Science, University of Mustansiriyah   82  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (2) 2020 𝑁𝑎𝑏 ⊆ 𝐾 where 𝑎, 𝑏 ∈ 𝑅 and 𝐾 is a submodule of 𝑀 implies either 𝑁𝑎 ⊆ 𝐾 or 𝑁𝑏 ⊆ 𝐾 for some positive integer 𝑡. A nonzero submodule 𝑁 is a secondary submodule of 𝑀 if for any 𝑎 ∈ 𝑅, the endomorphism 𝑓 : 𝑁 → 𝑁 defined by 𝑓 𝑛 𝑛𝑎 for each 𝑛 ∈ 𝑁, is either surjective or nilpotent ( that is 𝐼𝑚𝑓 𝑁𝑎 𝑁 or 𝐼𝑚𝑓 𝑁𝑎 0 for some positive integer 𝑡 ) [1]. Equivalently, 0 𝑁 is a secondary submodule of 𝑀 if for every ideal 𝐼 of 𝑅, 𝑁𝐼 𝑁 or 𝑁𝐼 0 for some positive integer 𝑡 [1]. In this case, 𝑎𝑛𝑛 𝑁 is a primary ideal of 𝑅 (that is 𝑎𝑛𝑛 𝑁 is a prime ideal of 𝑅) [1]. A proper submodule 𝐾 of 𝑀 is classical primary if 𝑁𝑎𝑏 ⊆ 𝐾 where 𝑎, 𝑏 ∈ 𝑅 and 𝑁 is a submodule of 𝑀 then 𝑁𝑎 ⊆ 𝐾 or 𝑁𝑏 ⊆ 𝐾 for some positive integer 𝑡 [6]. A proper submodule 𝐾 of 𝑀 is called completely irreducible when 𝐾 ⋂ 𝐻∈∧ where 𝐻 ∈∧ is a family of submodules of 𝑀 implies that 𝐾 𝐻 for some 𝑖 ∈∧ [2]. It is not hard to see that every submodule is an intersection of completely irreducible submodules of 𝑀 consequently the intersection of all completely irreducible submodules of 𝑀 is zero. 𝑁 is called simple (sometimes minimal) submodule of a module 𝑀 if 𝑁 0 and for each submodule 𝐿 of 𝑀 and 𝑁 contains 𝐿 properly implies 𝐿 0 [7]. 𝑀 is coquasi-dedekind if all nonzero endomorphism of 𝑀 is epimorphism (in other word, 𝑓 𝑀 𝑀 for every 0 𝑓 ∈ 𝑆 ) [8]. Let 𝑅 be a commutative integral domain, 𝑀 is called divisible module over 𝑅 if 𝑀𝑎 𝑀 for each 0 𝑎 ∈ 𝑅 [7]. A proper submodule 𝑁 is maximal if it is not properly contained in any proper submodule of 𝑀 [7]. A proper submodule 𝑁 is called prime if 𝑚𝑟 ∈ 𝑁 implies 𝑚 ∈ 𝑁 or 𝑀𝑟 ⊆ 𝑁 [9]. 𝑀 is called a prime module if the zero submodule is prime. A proper ideal 𝐼 is prime if 𝑎𝑏 ∈ 𝐼 where 𝑎, 𝑏 ∈ 𝑅 implies 𝑎 ∈ 𝐼 or 𝑏 ∈ 𝐼 [10]. Equivalently, a proper ideal 𝐼 is prime if 𝐴𝐵 ⊆ 𝐼 where 𝐴 and 𝐵 are ideals of 𝑅 implies 𝐴 ⊆ 𝐼 or 𝐵 ⊆ 𝐼 [10]. A ring in which every ideal prime is called fully prime [11]. Equivalently, a ring 𝑅 is fully prime if and only if it is fully idempotent (a ring in which every ideal is an idempotent that is 𝐼 𝐼 for each ideal 𝐼 of ) and the set of ideals of 𝑅 is totally ordered under inclusion [11]. A proper submodule 𝑁 is called primary if 𝑚𝑟 ∈ 𝑁 implies 𝑚 ∈ 𝑁 or 𝑀𝑟 ⊆ 𝑁 for some positive integer 𝑡 [6]. 𝑀 is called a primary module if the zero submodule is primary. A proper ideal 𝐼 is primary if 𝑎𝑏 ∈ 𝐼 where 𝑎, 𝑏 ∈ 𝑅 implies 𝑎 ∈ 𝐼 or 𝑏 ∈ 𝐼 for some positive integer 𝑡 [6]. 0 𝑀 is called an 𝑆-second module if for every 𝑓 ∈ 𝑆 implies 𝑓 𝑀 𝑀 or 𝑓 𝑀 0 [12]. 0 𝑀 is called an S-weakly second module whenever 𝑓𝑔 𝑀 ⊆ 𝐾, where 𝑓, 𝑔 ∈ 𝑆 and 𝐾 a submodule of 𝑀 implies either 𝑓 𝑀 ⊆ 𝐾 or 𝑔 𝑀 ⊆ 𝐾 [3]. Equivalently, 𝑀 is an S-weakly second module if and only if for each 𝜁, 𝜗 ∈ 𝑆 implies 𝜁𝜗 𝑀 𝜁 𝑀 or 𝜁𝜗 𝑀 ⊇ 𝜗 𝑀 [3]. 𝑀 is called multiplication when each submodule 𝑁 of 𝑀, we have 𝑁 𝑀𝐼 for some ideal 𝐼 of 𝑅 [13]. We able to take 𝐼 𝑁 : 𝑀 𝑟 ∈ 𝑅 and 𝑀𝑟 ⊆ 𝑁 is an ideal of 𝑅 [13]. 𝑀 is called faithful if 0: 𝑀 𝑎𝑛𝑛 𝑀 𝑟 ∈ 𝑅 and 𝑀𝑟 0 0. 𝑀 is a scalar module when for each 𝑓 ∈ 𝐸𝑛𝑑 𝑀 there is 𝑎 ∈ 𝑅 with 𝑓 𝑚 𝑚𝑎 for all 𝑚 ∈ 𝑀 [14]. The aim of this research is to continue studying the concept of semisecond submodules. A nonzero submodule 𝑁 of 𝑀 is called semisecond if for each 𝑎 ∈ 𝑅, 𝑁𝑎 𝑁𝑎 [2]. A nonzero module 𝑀 is said to be semisecond if 𝑀 is semisecond submodule of itself. In fact this idea is the dual notion of the concept semiprime submodules. A proper submodule of 𝑀 is called semiprime if for each 𝑎 ∈ 𝑅, 𝑚 ∈ 𝑀 such that 𝑚𝑎 ∈ 𝑁 implies 𝑚𝑎 ∈ 𝑁 [9]. A proper ideal 𝐼 of 𝑅 is semiprime if for each 𝑎 ∈ 𝑅 such that 𝑎 ∈ 𝐼 implies 𝑎 ∈ 𝐼 [7]. Equivalently, a proper ideal 𝐼 of 𝑅 is semiprime if for each ideal 𝐴 of 𝑅 such that 𝐴 ⊆ 𝐼 implies 𝐴 ⊆ 𝐼 [7]. It is well-known that 𝑅 is fully semiprime (that is 𝑅 in which every ideal is   83  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (2) 2020 semiprime ) if and only if 𝑅 is von Neumann regular ( that is for every 𝑎 ∈ 𝑅, there is 𝑏 ∈ 𝑅 such that 𝑎 𝑎𝑏𝑎 ) [15]. It is well-known if 𝑅 is commutative then 𝑅 is von Neumann regular if and only if 𝑎𝑅 𝑎 𝑅 if and only if every ideal of 𝑅 is pure ( that is 𝐼 ∩ 𝐽 𝐼𝐽 for each ideal 𝐼 and 𝐽 of 𝑅 ) if and only if 𝑅 is fully idempotent. And 𝑀 is called regular if for every 𝑚 ∈ 𝑀 and for every 𝑎 ∈ 𝑅 we have 𝑚𝑎 𝑚𝑎𝑟𝑎 for some 𝑟 ∈ 𝑅. If 𝑀 is regular then every submodule of 𝑀 is pure (that is every submodule 𝑁 of 𝑀 satisfying 𝑁𝐼 𝑀𝐼 ∩ 𝑁 for each ideal 𝐼 of 𝑅) [15]. If 𝑅 is commutative then 𝑀 is regular if and only if for every 𝑚 ∈ 𝑀 and for every 𝑎 ∈ 𝑅 we have 𝑚𝑎 𝑚𝑎 𝑟 for some 𝑟 ∈ 𝑅. Also 𝑅 is Boolean ring if 𝑎 𝑎 for every 𝑎 ∈ 𝑅 [7]. Thus a Boolean ring is von Neumann. We call a module 𝑀 is Rickart when for every 𝑓 ∈ 𝐸𝑛𝑑 𝑀 , 𝑘𝑒𝑟𝑓 is a direct summand of 𝑀 [16]. 𝑀 is a dual Rickart module when for every 𝑓 ∈ 𝐸𝑛𝑑 𝑀 , Im 𝑓 is a direct summand of 𝑀 [16]. It is well- known that for each 𝑎 ∈ 𝑅 we can define 𝑓 : 𝑅 → 𝑅 by 𝑓 𝑟 𝑎𝑟 for each 𝑟 ∈ 𝑅 then 𝐼𝑚𝑓 𝑎𝑅. This means 𝑅 is von Neumann regular if and only if 𝑅 is dual Rickart as 𝑅- module. A nonzero submodule 𝑁 of 𝑀 is weak semisecond whenever 𝑁𝑎 ⊆ 𝐾 where 𝑎 ∈ 𝑅 and 𝐾 a submodule of 𝑀 implies either 𝑁𝑎 ⊆ 𝐾 or 𝑎 ∈ 𝑎𝑛𝑛 𝑁 [17]. A nonzero submodule 𝑁 of 𝑀 is called a strongly 2-absorbing second submodule if for each 𝑎, 𝑏 ∈ 𝑅, we have 𝑁𝑎𝑏 𝑁𝑎 or 𝑁𝑎𝑏 𝑁𝑏 or 𝑁𝑎𝑏 0 [18]. A module 𝑀 is called cacellation if 𝑀𝐼 𝑀𝐽 implies 𝐼 𝐽 for each ideal 𝐼 and 𝐽 of 𝑅 [19]. Other works within [20-23]. Is related topics. The paper contains five branches and better say “sections”). In second part, we give other descriptions of the semisecond submodules idea (Theorem 2.2, Theorem 2.4, and Proposition 2.8). More examples and information about this idea are provided (Remarks and Examples 2.3). We study the homomorphic image and the direct sum of this class of modules (Proposition 2.5 and Propsition 2.6). Section three includes (Theorem 3.1) is the most important tool to describe semisecond submodules. More characterizations are supplied (Corollary 3.9 and Theorem 3.12). Section four is devoted to finding any relationships between semisecond submodules and related modules. Among other observations, we see that every nonzero regular module over a commutative ring is semisecond (Theorem 4.1). The semisecond and von Neumann regular concepts are coincident in the commutative rings (Theorem 4.7). In section five, we present the concept S-semisecond submodules and the basic properties of this modules is investigated.    In what follows, ℤ, ℚ, ℤ , ℤ ℤ ℤ and 𝑀𝑎𝑡 𝑅 we denote respectively, integers, rational numbers, the 𝑝-Prüfer group, the residue ring modulo 𝑛 and an 𝑛 𝑛 matrix ring over 𝑅 . 2. Semisecond Submodules We give a characterization of semisecond submodules, first we recall the main definition. Definition (2.1) [2]. A nonzero submodule 𝑁 of 𝑅-module 𝑀 is called semisecond if 𝑁𝑎 𝑁𝑎 for each 𝑎 ∈ 𝑅. Theorem (2.2): The following assertions are equivalent (1) 𝑁 is a semisecond submodule of an 𝑅-module 𝑀 (2) 𝑁 0 and whenever 𝑁𝑎 ⊆ 𝐾, where 𝑎 ∈ 𝑅 and 𝐾 a submodule of 𝑀 implies 𝑁𝑎 ⊆ 𝐾   84  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (2) 2020 Proof. (1)  (2) Let 𝑎 ∈ 𝑅 and 𝐾 a submodule of 𝑀 with 𝑁𝑎 ⊆ 𝐾. Because 𝑁 is semisecond then 𝑁 0 and 𝑁𝑎 𝑁𝑎 for each 𝑎 ∈ 𝑅 implies 𝑁𝑎 𝑁𝑎 ⊆ 𝐾 as desired. (3)  (1) Assume 𝑁 0 and 𝑎 ∈ 𝑅 then 𝑁𝑎 ⊆ 𝑁𝑎 . By hypothesis 𝑁𝑎 ⊆ 𝑁𝑎 and hence 𝑁𝑎 𝑁𝑎 as required. Remarks and Examples (2.3) (1) Obviously semisecond submodules are weak semisecond but the converse fails for more information see [17]. (2) It is clear that weakly second submodules are semisecond. The converse is not hold in general, ℤ as ℤ-module is semisecond since ℤ . 𝑎 ℤ . 𝑎 for each 𝑎 ∈ ℤ but ℤ is not weakly second because ℤ . 3 ℤ . 2.3 0 ℤ . 2. (3) As another example of (2), let 𝑁 ℤ ⊕ ℤ be a submodule of 𝑀 ℤ ⊕ ℤ as ℤ-module where 𝑝 and 𝑞 prime numbers. Then 𝑁 is semisecond since 𝑁𝑎 𝑁𝑎 for each 𝑎 ∈ ℤ but 𝑁 is not a weakly second submodule of 𝑀 because 𝑁. 𝑝. 𝑞 0 while 𝑁. 𝑝 0 ⊕ ℤ and 𝑁. 𝑞 ℤ ⊕ 0. (4) Clearly every module over Boolean ring is semisecond. (5) Secondary and weakly secondary submodules not necessarily semisecond. Consider ℤ as ℤ-module is secondary (and hence weakly secondary) see [4]. But 𝑀 is not semisecond because ℤ . 2 ℤ . 2 . (6) Semisecond submodules also need not be secondary or weakly secondary submodules. For example: ℤ as ℤ-module is semisecond by (2) but ℤ is not weakly secondary and hence it is not secondary see [4]. (7) It is obvious that coquasi-dedekind (or simple or divisible) submodule  second submodule  strongly 2-absorbing second submodules  weakly second submodules  semisecond submodules  weak semisecond submodules. The converse is not true in general, 𝑀 ℤ ⊕ ℤ as ℤ-module is semisecond but it is not strongly 2- absorbing second,(and hence not weakly second ) since 𝑀. 3 𝑀2.3 0 ⊕ ℤ 𝑀. 2 and 𝑀. 2.3 0 . (8) If 𝑁 is a maximal (and hence prime ) submodule then 𝑁 may not be semisecond. For example, 𝑁 𝑝ℤ is a maximal submodule of ℤ as ℤ-module but 𝑁 is not semisecond since 𝑁𝑎 𝑁𝑎 for every 𝑎 ∈ ℤ and any prime number 𝑝. (9) Let 𝑁and 𝐻 be submodules of an 𝑅-module 𝑀 with 𝑁 ⊆ 𝐻 ⊆ 𝑀. If 𝑁 is a smisecond submodule of 𝑀 then 𝐻 needs not be a semisecond submodule of 𝑀. Let 𝑁 ℤ . 2 and 𝐻 ℤ 𝑀 submodules of 𝑀 ℤ as ℤ-module where 𝑁 is a simple submodule so it is semisecond while 𝐻 is not semisecond by (5). (10) Let 𝑁 and 𝐻 be submodules of an 𝑅-module 𝑀 with 𝑁 ⊆ 𝐻 ⊆ 𝑀. If 𝐻 is a sermisecond submodule of , then 𝑁 needs not be a semisecond submodule of 𝑀. Let 𝑁 ℤ be a submodule of 𝑀 ℤ as ℤ-module. Since 𝑀 is a divisible module then 𝑀 is semisecond but 𝑁 is not semisecond because 𝑁. 𝑝 0 𝑁𝑝 ℤ . (11) As another example of (10), ℚ as ℤ-module is divisible so it is semisecond but the submodule ℤ is not semisecond.   85  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (2) 2020 Theorem (2.4): The following assertions are equiavalent (1) 𝑁 is a semisecond submodule of an 𝑅-module 𝑀. (2) 𝑁 0 and for each 𝑎, 𝑏 ∈ 𝑅 and 𝐾 a finite intersection of completely irreducible submodules of 𝑀 with 𝑁𝑎 ⊆ 𝐾 implies 𝑁𝑎 ⊆ 𝐾. Proof. (1)  (2) it is clear. (3)  (1) Let 0 𝑁 and 𝐾 are submodules of 𝑀 with 𝑁𝑎 ⊆ 𝐾 where 𝑎 ∈ 𝑅. Suppose 𝑁𝑎 ⊈ 𝐾 implies 𝐾 ∩ ∈∧ 𝐻 for some collection 𝐻 ∈∧ of completely irreducible submodules of 𝑀. We have 𝑁𝑎 ⊈∩ ∈∧ 𝐻 . So there exists 𝑖 ∈∧ such that 𝑁𝑎 ⊈ 𝐻 . On the other hand, 𝑁𝑎 ⊆ 𝐾 ∩ ∈∧ 𝐻 and hence 𝑁𝑎 ⊆ 𝐾 ⊆ ⋂ 𝐻 for some positive integer 𝑛 because 𝐾 ⊆ 𝐻 for each 𝑖 ∈∧. By hypothesis, 𝑁𝑎 ⊆ ⋂ 𝐻 .Then 𝑁𝑎 ⊆ 𝐻   which is a contradiction as required. Proposition (2.5): Every nonzero homomorphic image of semisecond submodule is smisecond. Proof. Let 𝐴 and 𝐵 be 𝑅-modules and 0 𝑓: 𝐴 → 𝐵 an 𝑅-homomorphism. Let 𝑁 be a semisecond submodule of 𝐴. Firstly, since 𝑓 0 implies 𝑓 𝑁 0. For each 𝑎 ∈ 𝑅 then 𝑓 𝑁 𝑎 𝑓 𝑁𝑎 𝑓 𝑁𝑎 𝑓 𝑁 𝑎 . Proposition (2.6): Let 𝑁 and 𝑁 be non-zero submodules of 𝑀 and 𝑀 𝑅-modules respectively. Then 𝑁 𝑁 ⊕ 𝑁 is a semisecond submodule of 𝑀 𝑀 ⊕ 𝑀 if and only if 𝑁 and 𝑁 are semisecond submodules of 𝑀 and 𝑀 respectively. Proof. () Let 𝑎 ∈ 𝑅 then 𝑁 ⊕ 𝑁 𝑎 𝑁 ⊕ 𝑁 𝑎 and hence 𝑁 𝑎 ⊕ 𝑁 𝑎 𝑁 𝑎 ⊕ 𝑁 𝑎 implies 𝑁 𝑎 𝑁 𝑎 and 𝑁 𝑎 𝑁 𝑎 as required. () it is clear. Corollary (2.7): Every non-zero direct summand of a semisecond module is semisecond. Proposition (2.8): The following statements are equivalent (1) 𝑁 is a semisecond submodule of 𝑅-module 𝑀. (2) is a semisecond submodule of 𝑅-module for each submodule 𝐻 of 𝑀 contained in 𝑁. Proof. (1)  (2) Let 𝑁 be a semisecond submodule 𝑀 and 𝜋: 𝑀 → be the natural homomorphism for each submodule 𝐻 of 𝑀 contained in 𝑁 so by Proposition 2.5, 𝜋 𝑁 is a semisecond submodule . (2)  (1) It is clear by taking 𝐻 0. 3. More Characterizations and Facts About Semisecond Submodules Theorem (3.1): The following statements are equivalent (1) 𝑁 is a semisecond submodule of an 𝑅-module 𝑀. (2) 𝑁 0 and 𝐾: 𝑁 is a semiprime ideal of 𝑅 for each submodule 𝐾 ⊉ 𝑁 in 𝑀. Proof. (1)  (2) Assume 𝑁 is a semisecond submodule of an 𝑅-module 𝑀 and 𝐾 a submodule of 𝑀 such that 𝑁 ⊈ 𝐾 implies 𝐾: 𝑁 𝑅. Let 𝑎 ∈ 𝑅 with 𝑎 ∈ 𝐾: 𝑁 implies 𝑁𝑎 ⊆ 𝐾 thus 𝑁𝑎 ⊆ 𝐾 and hence 𝑎 ∈ 𝐾: 𝑁 as required.   86  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (2) 2020 (2)  (1) Let 𝑁 and 𝐾 be submodules of an 𝑅-module 𝑀 such that 𝑁𝑎 ⊆ 𝐾 where 𝑎 ∈ 𝑅. In case 𝑁 ⊆ 𝐾 then already 𝑁𝑎 ⊆ 𝐾. If 𝑁 ⊈ 𝐾 then 𝐾: 𝑁 is a semiprime ideal of 𝑅 by hypothesis and 𝑎 ∈ 𝐾: 𝑁 implies 𝑁𝑎 ⊆ 𝐾 as desired. Corollary (3.2): Every submodule of a module over a fully semiprime (that is von Neumann regular) ring is semisecond. Proof. Directly via Theorem 3.1. Corollary (3.3): If 𝑁 is a semisecond submodule of an 𝑅-module 𝑀 then 𝑎𝑛𝑛 𝑁 is a semiprime ideal of 𝑅. Proof. Directly via Theorem 3.1. Examples (3.4): 𝑎𝑛𝑛 𝑁 0 is a semiprime ideal of ℤ for every nonzero submodule 𝑁 of the ℤ-module ℤ while 𝑁 is not semisecond. Corollary (3.5): If 𝑁 is a semisecond submodule of an 𝑅-module 𝑀 then for every submodule 𝐾 ⊉ 𝑁 in 𝑀 we have 𝐾: 𝑁 𝐾: 𝑁𝑏 for each 𝑏 ∈ 𝑅. Proof. Let 𝑎 ∈ 𝐾: 𝑁 then 𝑁𝑎 ⊆ 𝐾 implies for each 𝑏 ∈ 𝑅 𝑁𝑎𝑏 ⊆ 𝐾 so 𝑎 ∈ 𝐾: 𝑁𝑏 . Conversly, let 𝑎 ∈ 𝐾: 𝑁𝑏 then 𝑁𝑎𝑏 ⊆ 𝐾 implies 𝑎𝑏 ∈ 𝐾: 𝑁 and we can take 𝑏 𝑎 then 𝑎 ∈ 𝐾: 𝑁 . Via Theorem 3.1, 𝐾: 𝑁 is a semiprime ideal of 𝑅 implies 𝑎 ∈ 𝐾: 𝑁 as required. Corollary (3.6): If 𝑁 is a semisecond submodule of an 𝑅-module 𝑀 then 𝑎𝑛𝑛 𝑁 𝑎𝑛𝑛 𝑁𝑏 for each 𝑏 ∈ 𝑅. Proof. Directly by Corollary 3.5. Theorem (3.7): The following statements are equivalent (1) 𝑁 is a semisecond submodule of an 𝑅-module 𝑀. (2) 𝑁 0 and for each ideals 𝐼 of 𝑅 such that 𝑁𝐼 ⊆ 𝐾 implies 𝑁𝐼 ⊆ 𝐾. Proof. (1)  (2) First since 𝑁 is a semisecond submodule of an 𝑅-module 𝑀 then 𝑁 0. Let 𝐼 be an ideal of 𝑅 and 𝐾 a submodule of 𝑀. If 𝑁 ⊈ 𝐾 we have either 𝑁𝐼 ⊈ 𝐾 and so nothing to prove or 𝑁𝐼 ⊆ 𝐾 it follows 𝐼 ⊆ 𝐾: 𝑁 and by Theorem 3.1, 𝐾: 𝑁 is a semiprime ideal of 𝑅 so 𝐼 ⊆ 𝐾: 𝑁 and hence 𝑁𝐼 ⊆ 𝐾. In case 𝑁 ⊆ 𝐾 then the result already is obtained. (2)  (1) Let 𝑁𝑎 ⊆ 𝐾, where 𝑎 ∈ 𝑅 and 𝐾 a submodule of 𝑀, then 𝑁 𝑎 ⊆ 𝐾. By hypothesis 𝑁 𝑎 ⊆ 𝐾 where 𝑎 is the principal ideal generated by 𝑎 and hence 𝑁𝑎 ⊆ 𝐾 as dsired. Corollary (3.8): The following statements are equivalent (1) 𝑁 is a semisecond submodule of an 𝑅-module 𝑀. (2) 𝑁 0 and for each ideal 𝐼 of 𝑅 and 𝐾 a submodule of 𝑀 such that 𝑁 ⊈ 𝐾 and 𝐼 ⊆ 𝐾: 𝑁 implies 𝐼 ⊆ 𝐾: 𝑁 . Proof. Directly via corollary 3.7. Corollary (3.9): The following statements are equivalent (1) 𝑁 is a semisecond submodule of an 𝑅-module 𝑀. (2) 𝑁 0 and for each ideal 𝐼 of 𝑅 implies 𝑁𝐼 𝑁𝐼. Proof. (1)  (2) First since 𝑁 is a semisecond submodule of an 𝑅-module 𝑀 then 𝑁 0. Let 𝐼 be an ideal of 𝑅 then 𝑁𝐼 ⊆ 𝑁𝐼 so by Theorem 3.7, we have 𝑁𝐼 ⊆ 𝑁𝐼 and thus 𝑁𝐼 𝑁𝐼.   87  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (2) 2020 (2)  (1) it is clear. Theorem (3.10): Let 𝑁 be a submodule of an 𝑅-module 𝑀. If for each 𝑎 ∈ 𝑅, 𝑎 𝑅 𝑎𝑛𝑛 𝑁 𝑎𝑅 𝑎𝑛𝑛 𝑁 then 𝑁 is semisecond. Proof. Assume for each 𝑎 ∈ 𝑅, 𝑎 𝑅 𝑎𝑛𝑛 𝑁 𝑎𝑅 𝑎𝑛𝑛 𝑁 then 𝑎 𝑏 𝑎 𝑐 for some 𝑏, 𝑐 ∈ 𝑎𝑛𝑛 𝑁 implies 𝑎 𝑎 ∈ 𝑎𝑛𝑛 𝑁 and hence 𝑁𝑎 𝑁𝑎. Theorem (3.11): If 𝑁 is a semisecond finitely generated submodule of an 𝑅-module 𝑀 then for each 𝑎 ∈ 𝑅, 𝑎 𝑅 𝑎𝑛𝑛 𝑁 𝑎𝑅 𝑎𝑛𝑛 𝑁 . Proof. Let 𝑎 ∈ 𝑅 then 𝑁𝑎 𝑁𝑎 that is 𝑁 𝑎𝑅 𝑎𝑅 𝑁 𝑎𝑅 . By hypothesis 𝑁 is finitely generated. It is not hard to see that 𝑁 𝑎𝑅 is also finitely generated. Via [23, Corollary 2.5], it follows that 𝑥 1 ∈ 𝑅𝑎 and 𝑁𝑥 𝑎𝑅 0. Let 𝑥 1 𝑎𝑡 for some 𝑡 ∈ 𝑅 then 𝑥 𝑎𝑡 1 implies 𝑁 𝑎𝑡 1 𝑎 0. This means 𝑎 𝑡 𝑎 ∈ 𝑎𝑛𝑛 𝑁 so 𝑎 𝑡 𝑎 𝑏 for some 𝑏 ∈ 𝑎𝑛𝑛 𝑁 implies 𝑎 𝑎 𝑡 𝑏 and hence 𝑎𝑅 𝑎𝑛𝑛 𝑁 ⊆ 𝑎 𝑅 𝑎𝑛𝑛 𝑁 . Then 𝑎 𝑅 𝑎𝑛𝑛 𝑁 𝑎𝑅 𝑎𝑛𝑛 𝑁 . Theorem (3.12): Let 𝑁 be a finitely generated submodule of a module 𝑀 over a commutative ring 𝑅. The following statements are equivalent (1) 𝑁 is semisecond. (2) For each 𝑎 ∈ 𝑅, 𝑁𝑎 𝑁𝑟 𝑁𝑟 for some 𝑟 ∈ 𝑅. Proof. (1)  (2) By Theorem 3.11, for each 𝑎 ∈ 𝑅, 𝑎 𝑅 𝑎𝑛𝑛 𝑁 𝑎𝑅 𝑎𝑛𝑛 𝑁 . Then 𝑎 𝑡 𝑏 𝑎𝑠 𝑐 for some 𝑠, 𝑡 ∈ 𝑅 and 𝑏, 𝑐 ∈ 𝑎𝑛𝑛 𝑁 . By choosing 𝑠 1 we have 𝑎 𝑎 𝑡 𝑑 for some 𝑑 𝑏 𝑐 ∈ 𝑎𝑛𝑛 𝑁 thus 𝑎𝑅 ⊆ 𝑎𝑡𝑅 𝑎𝑛𝑛 𝑁 implies 𝑎𝑅 𝑎𝑛𝑛 𝑁 ⊆ 𝑎𝑡𝑅 𝑎𝑛𝑛 𝑁 hence 𝑎𝑅 𝑎𝑛𝑛 𝑁 𝑎𝑡𝑅 𝑎𝑛𝑛 𝑁 . Put 𝑟 𝑎𝑡 it follows 𝑎 𝑟 ∈ 𝑎𝑛𝑛 𝑁 . Therefore 𝑁𝑎 𝑁𝑟 but 𝑎𝑡 𝑎 𝑡 𝑑𝑡 that is 𝑟 𝑟 ∈ 𝑎𝑛𝑛 𝑁 thus 𝑁𝑎 𝑁𝑟 𝑁𝑟 as desired. (2)  (1) for each 𝑎 ∈ 𝑅, 𝑁𝑎 𝑁𝑎𝑎 𝑁𝑟𝑎 𝑁𝑎𝑟 𝑁𝑟𝑟 𝑁𝑟 𝑁𝑎 implies 𝑁 is semisecond. 4. Semisecond Submodules and Related Concepts Let us start by the following observation (observation) Theorem (4.1): Every non-zero regular module over a commutative ring is semisecond. Proof. Let 𝑀 be a nonzero regular 𝑅-module. We show 𝑀𝑎 𝑀𝑎 for each 𝑎 ∈ 𝑅. Let 𝑥 ∈ 𝑀𝑎 implies 𝑥 𝑚𝑎 for some 𝑚 ∈ 𝑀 it follows 𝑚𝑎 𝑚𝑎𝑟𝑎 𝑚𝑎 𝑟 ∈ 𝑀𝑎 for some 𝑟 ∈ 𝑅. Example (4.2): (1) Every regular ideal 𝐼 of commutative ring 𝑅 is a semisecond as 𝑅-module. (2) ℤ and ℚ as ℤ-modules are semisecond but not regular. Corollary (4.3): Every non-zero module over commutative von Neumann regular ring is semisecond. Proof. Since every module over von Neumann regular ring is regular so the result follows by Theorem 4.1. Corollary (4.4): Every nonzero submodule of a regular module over commutative ring is semisecond. Proof. Since every submodule of a regular module is regular, so by theorem 4.1 we already have the result.   88  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (2) 2020 Corollary (4.5): Every nonzero semisimple module over commutative ring is semisecond. Corollary (4.6): Every submodule of a semisimple module over commutative ring is semisecond. Theorem (4.7): The von Neumann regular and semisecond notions in the commutative rings are the same. Proof. It is clear by definitions both notions. Examples (4.8): (1) The commutativity condition in Theorem and Theorem cannot be dropped. Consider the ring 𝑅 ℤ ℤ ℤ ℤ as a right 𝑅-module. By simple calculation, we see that 𝑅 is von Neumann regular and 𝑅 is not commutative. On the other hand, if we take 𝑎 0 1 0 0 ∈ 𝑅 implies 𝑎𝑅 𝑎 𝑅 0 0 0 0 , what follows 𝑅 is not a semisecond ring. (2) Consider the ring 𝑅 ℤ ℤ 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 0 0 as a right 𝑅- module where 𝑅 is not commutative. By simple steps, we have 𝑎𝑅 𝑎 𝑅 for each 𝑎 ∈ 𝑅 it follows that 𝑅 is semisecond but 𝑅 is not von Neumann regular since 0 1 0 0 0 1 0 0 𝑏 0 1 0 0 for each 𝑏 ∈ 𝑅. (3) Semisecond modules may not be semisimple. Consider 𝑅 ∏ 𝔽∈∧ is commutative von Neumann regular ring ( 𝑅 is a regular as 𝑅-module ) and hence 𝑅 is semisecond but 𝑅 is not semisimple since the submodule 𝑅 ⊕ ∈∧ 𝔽 is not a direct summand of 𝑅. Proposition (4.9): Let 𝑅 be a commutative ring then we have the equivalent (1) 𝑅 is von Neumann regular. (2) 𝑅 is fully semiprime. (3) 𝑅 is fully idempotent. (4) 𝑅 is a dual Rickart as 𝑅-module. (5) 𝑅 is semisecond (6) 𝑅 is cosemisimple. Proof. (1)  (2)  (3)  (4) as we mentioned before where the commutativity condition is not necessary, (1)  (5) by Theorem 4.4 and (1)  (6) via [7]. Proposition (4.10): Every nonzero module over semisecond ring is semisecond. Proof. Let 0 𝑀 be a module over a semisecond ring 𝑅 implies 𝑅𝑎 𝑅𝑎 and thus 𝑀𝑎 𝑀𝑎. Example (4.11): Let 𝑅 ℤ ℤ 0 ℤ and 𝑀 0 ℤ 0 ℤ be considered as a right 𝑅- module. By simple steps, we see that 𝑀𝑎 𝑀𝑎 for each 𝑎 ∈ 𝑅 that 𝑀 is semisecond but 𝑅 is not semisecond since if we take 𝑎 0 1 0 0 we have 𝑅𝑎 𝑅𝑎 . In fact if 𝑅 is semisecond, then 𝑀 is semisecond which is a contradiction by Proposition 4.3. Moreover, 𝑀 is not semisimple since 0 ℤ 0 0 is a cyclic submodule of 𝑀 which is not a direct summand   89  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (2) 2020 of 𝑀. Also, 𝑀 is not regular since 0 ℤ 0 ℤ 0 0 0 1 ∩ 0 ℤ 0 0 0 ℤ 0 0 0 ℤ 0 0 0 0 0 1 0 0 0 0 thus 0 ℤ 0 0 is not a pure submodule of 𝑀. Corollary (4.12): Let 𝑀 be an 𝑅-module and 𝐼 be an ideal of 𝑅 such 𝐼 ⊆ 𝑎𝑛𝑛 𝑀 . If is a semisecond ring then 𝑁 is semisecond. Proof. Since 𝑁 is considered as –module so by Proposition 4.10, the result is obtained. Proposition (4.13): Let 𝑀 be an 𝑅-module and 𝐼 be an ideal of 𝑅 such that 𝐼 ⊆ 𝑎𝑛𝑛 𝑀 . Then 𝑀 is a semisecond 𝑅-module if and only if 𝑀 is a semisecond -module. Proof. It is clear. Examples (4.14): (1) ℤ and ℚ as ℤ-modules are semisecond but ℤ ℤ ℚ ≅ ℤ ≅ ℤ ℤ ℤ is not semisecond. (2) Consider ℤ as ℤ-module implies ℤ ℤ ℤ ℤ is semisecond but 𝑅 ℤ is not semisecond. Proposition (4.15): If 𝑁 is a cancellation semisecond submodule of an 𝑅-module 𝑀 then 𝑅 is semisecond. Proof. For each 𝑎 ∈ 𝑅, we have 𝑎 𝑁 𝑎𝑁, then 𝑎 𝑅 𝑁 𝑎𝑅 𝑁 and since 𝑁 is cancellation implies 𝑎 𝑅 𝑎𝑅 as desired. Corollary (4.16): If 𝑀 is a finitely generated faithful multiplication semisecond 𝑅-module then 𝑅 is fully idempotent (and hence semisecond). Proof. Let 𝑀 be a semisecond 𝑅-module then 𝐼 𝑀 𝐼𝑀 for each ideal 𝐼 of 𝑅. Since 𝑀 is a finitely generated faithful multiplication so by [13], 𝑀 is cancellation then 𝐼 𝐼 thus 𝑅 is fully idempotent. Corollary (4.17): If 𝑀 is a cancellation (or finitely generated faithful multiplication) semisecond 𝑅-module such that the set of ideals of 𝑅 is totally ordered under inclusion then 𝑅 is fully prime. Proof. By Corollary 4.15, 𝑅 is fully idempotent so by [11]. 𝑅 is fully prime. Theorem (4.18): Let 𝑀 be a multiplication 𝑅-module. If 𝑁: 𝑀 is a semisecond ideal of 𝑅 then 𝑁 is a semisecond submodule of 𝑀. Proof. By hypothesis, 𝑁: 𝑀 𝐼 𝑁: 𝑀 𝐼 for each ideal 𝐼 of 𝑅 then 𝑀 𝑁: 𝑀 𝐼 𝑀 𝑁: 𝑀 𝐼 . By hypothesis 𝑀 is multiplication thus 𝑁𝐼 𝑁𝐼 so 𝑁 is semisecond. Theorem (4.19): Let 𝑀 be a finitely generated faithful multiplication 𝑅-module. If 𝑁 is a semisecond submodule of 𝑀 then 𝑁: 𝑀 is a semisecond ideal of 𝑅. Proof. Since 𝑁𝐼 𝑁𝐼 for each ideal 𝐼 of 𝑅 then 𝑀 𝑁: 𝑀 𝐼 𝑀 𝑁: 𝑀 𝐼 because 𝑀 is multiplication. But 𝑀 is finitely generated faithful implies 𝑀 is cancellation and hence 𝑁: 𝑀 𝐼 𝑁: 𝑀 𝐼 thus 𝑁: 𝑀 is semisecond. Remark (4.20): If 𝐼 is a semisecond ideal of 𝑅 then 𝐼 𝐼 . Proof. Since 𝐼𝐽 𝐼𝐽 for each ideal 𝐽of 𝑅 so if we choose 𝐽 𝐼 implies 𝐼 𝐼 . Proposition (4.21): Every nonzero pure submodule of a semisecond module is semisecond.   90  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (2) 2020 Proof. Let 𝑁 be a nonzero pure submodule of a semisecond 𝑅-module 𝑀. Then for each ideal 𝐼 of 𝑅 implies 𝑁𝐼 𝑁 ∩ 𝑀𝐼 𝑁 ∩ 𝑀𝐼 𝑁𝐼 as desired. The following result is appeared in [2]. Without proof Proposition (4.22): Every sum of second submodules is semisecond. Proof. Let 𝑎 ∈ 𝑅, 𝑁 and 𝐻 be second submodules of an 𝑅-module 𝑀 implies either 𝑁 𝐻 𝑎 𝑁𝑎 𝐻𝑎 𝑁𝑎 𝐻𝑎 𝑁 𝐻 𝑎 or 𝑁 𝐻 𝑎 𝑁𝑎 𝐻𝑎 0 𝐻𝑎 𝐻𝑎 ⊆ 𝑁 𝐻 𝑎 or 𝑁 𝐻 𝑎 𝑁𝑎 𝐻𝑎 0 0 0 ⊆ 𝑁 𝐻 𝑎 and hence 𝑁 𝐻 𝑎 𝑁 𝐻 𝑎 . Example (4.23): The sum of second submodules may not be second. The submodules ℤ . 2 and ℤ . 3 are simple and hence second of ℤ as ℤ-module while ℤ . 2 ℤ . 3 ℤ is semisecond but not second. Proposition (4.24): Every semisecond submodule of prime module is second. Proof. Let 𝑎 ∈ 𝑅 and 𝑁 be a semisecond submodule of a prime 𝑅-module 𝑀 implies 𝑁𝑎 𝑁𝑎 then for each 𝑛 ∈ 𝑁 we have 𝑛𝑎 𝑚𝑎 for some 𝑚 ∈ 𝑁 implies 𝑛 𝑚𝑎 𝑎 0. But 0 is a prime submodule in, it follows either 𝑛 𝑚𝑎 ∈ 0 implies 𝑛 𝑚𝑎 and hence 𝑁 𝑁𝑎 or 𝑎 ∈ 0 : 𝑀 ⊆ 0 : 𝑁 implies 𝑁𝑎 0 as desired. Proposition (4.25): Every semisecond submodule of primary module is secondary. Proof. Similarly of Proposition 4.24. 5. 𝑺-Semisecond Modules Definition (5.1): A nonzero 𝑅-module 𝑀 is called 𝑆-semisecond whenever 𝑓 𝑀 ⊆ 𝐾, where 𝑓 ∈ 𝑆 𝐸𝑛𝑑 𝑀 and 𝐾 a submodule of 𝑀 implies 𝑓 𝑀 ⊆ 𝐾. Theorem (5.2): The following are equivalent (1) 𝑀 is a 𝑆-semisecond 𝑅-module. (2) 𝑀 0 and 𝑓 𝑀 𝑓 𝑀 for each 𝑓 ∈ 𝑆. Proof. (1)  (2) Assume 𝑀 is an 𝑆-semisecond 𝑅-module implies 𝑀 0. Since 𝑓 𝑀 ⊆ 𝑓 𝑀 implies 𝑓 𝑀 ⊆ 𝑓 𝑀 and hence 𝑓 𝑀 𝑓 𝑀 for each 𝑓 ∈ 𝑆 as desired. as desired. (2)  (1) Assume 𝑓 𝑀 ⊆ 𝐾, where 𝑓 ∈ 𝑆 and 𝐾 a submodule of 𝑀 implies 𝑓 𝑀 𝑓 𝑀 ⊆ 𝐾 as required. Proposition (5.3): Every semisecond multiplication module is 𝑆-semisecond. Proof. Let 𝑀 be a semisecond multiplication 𝑅-module and 𝑓 ∈ 𝑆 with 𝑓 𝑀 ⊆ 𝐾 for some 𝐾 a submodule of 𝑀. Since 𝑀 is multiplication then 𝑓 𝑀 𝑓 𝐼𝑀 𝐼𝑓 𝑀 𝐼𝐼𝑀 for some ideal 𝐼 of 𝑅 and hence 𝐼 𝑀 ⊆ 𝐾. By Theorem 3.6, we have 𝐼𝑀 ⊆ 𝐾 it follows 𝑓 𝑀 ⊆ 𝐾 that is 𝑀 is 𝑆-semisecond. Corollary (5.4): Every semisecond cyclic module is 𝑆-semisecond. Remarks and Examples (5.5): (1) Every 𝑆-semisecond module is semisecond. Proof. Let 𝑀 be an 𝑆-semisecond 𝑅-module, then 𝑀 0. Let 𝑀𝑎 ⊆ 𝐾 for some 𝑎 ∈ 𝑅 and 𝐾 a submodule of 𝑀. Define the endomorphisms 𝑓 : 𝑀 → 𝑀 by 𝑓 𝑚 𝑚𝑎 for each 𝑚 ∈ 𝑀. Then, 𝑓 𝑀 𝑓 𝑓 𝑀 𝑓 𝑀𝑎 𝑓 𝑀 𝑎 𝑀𝑎 ⊆ 𝐾. By hypothesis, we have 𝑓 𝑀 ⊆ 𝐾 that is 𝑀𝑎 ⊆ 𝐾 as desired. (2) The converse of (1) is not true in general. For example, 𝑀 ℤ ⊕ ℤ as ℤ- module is semisecond where   91  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (2) 2020 (3) 𝑆 𝐸𝑛𝑑ℤ ℤ ⊕ ℤ 𝐸𝑛𝑑ℤ ℤ 𝐻𝑜𝑚ℤ ℤ , ℤ 𝐻𝑜𝑚ℤ ℤ , ℤ 𝐸𝑛𝑑ℤ ℤ ≅ 𝑀𝑎𝑡 ℤ ℤ ℤ ℤ ℤ is not semisecond ring by Example 4.8(1) and ℤ ⊕ ℤ ≅ , , , so if we take 𝑓 0 1 0 0 ∈ 𝑆 𝐸𝑛𝑑ℤ ℤ ⊕ ℤ implies 𝑓 𝑀 0 1 0 0 , 𝑥, 𝑦 ∈ ℤ , 𝑓 𝑀 0 0 0 0 , 𝑥, 𝑦 ∈ ℤ it follows that 𝑀 is not semisecond as 𝑆-module that is, 𝑀 is not 𝑆-semisecond as ℤ-module. (4) If 0 𝑀 is not a divisible ℤ-module, then 𝑀 ⊕ 𝑀 can not be an 𝑆-semisecond ℤ- module. Proof. Let 𝑀 be a not divisible ℤ-module. Suppose that 𝑀 ⊕ 𝑀 is an 𝑆-semisecond ℤ-module. We can define the maps 𝑓: 𝑀 ⊕ 𝑀 → 𝑀 ⊕ 𝑀 𝑓 𝑥, 𝑦 𝑦2, 𝑥 for each 𝑥, 𝑦 ∈ 𝑀. It is clear that 𝑓 ∈ 𝑆 implies 𝑓 𝑀 ⊕ 𝑀 𝑓𝑓 𝑀 ⊕ 𝑀 𝑓 𝑀2 ⊕ 𝑀 𝑀2 ⊕ 𝑀2 𝑓 𝑀 ⊕ 𝑀 𝑀2 ⊕ 𝑀 which is a contradiction. (5) As another example of the converse of (1), we have ℤ ⊕ ℤ , ℤ ⊕ ℤ as ℤ-modules are semisecond but they are not 𝑆-semisecond by (2). In fact, 𝑆 is not semisecond ring so any module over 𝑆 cannot be semisecond by Proposition 4.10 as we mentioned in Proposition. (6) The direct sum of 𝑆-semisecond modules needs not be 𝑆-semisecond. For example, ℤ ⊕ ℤ , ℤ ⊕ ℤ are not 𝑆-semisecond as ℤ-modules (7) It is clear every that 𝑆-weakly second module is 𝑆-semisecond. The converse is not hold in general, ℤ as ℤ-module is 𝑆-semisecond since ℤ is multiplication and semisecond and hence it is 𝑆-semisecond but not weakly second and hence not 𝑆- weakly second. (8) As another example of (6), consider 𝑀 ℚ ⊕ ℤ as ℤ-module. Then 𝑆 𝐸𝑛𝑑ℤ ℚ ⊕ ℤ ≅ 𝐸𝑛𝑑ℤ ℚ 𝐻𝑜𝑚ℤ ℤ , ℚ 𝐻𝑜𝑚ℤ ℚ, ℤ 𝐸𝑛𝑑ℤ ℤ ℚ 0 0 ℤ is a commutative von Neumann regular ring and hence 𝑆 is semisecond so by Proposition 4.10, ℚ ⊕ ℤ is semisecond as 𝑆-module; that is, ℚ ⊕ ℤ is 𝑆-semisecond as ℤ-module. But ℚ ⊕ ℤ is not 𝑆-weakly second as ℤ-module since if we take 𝑓 1 0 0 0 , 𝑔 0 0 0 1 then 0 ⊕ ℤ 𝑔 𝑀 𝑓𝑔 𝑀 1 0 0 0 0 0 0 1 𝑥 𝑦  𝑥 ∈ ℚ, 𝑦 ∈ ℤ 0 0 0 0 𝑓 𝑀 ℚ ⊕ 0 (9) We have the implication Coquasi-dedekind modules  𝑆-second modules  𝑆-weakly second modules  𝑆-semisecond modules. Proposition (5.6): Every semisecond scalar module is 𝑆-semisecond. Proof. Let 𝑀 be a semisecond scalar 𝑅-module and 𝑓 ∈ 𝑆 with 𝑓 𝑀 ⊆ 𝐾 for some 𝐾 a submodule of 𝑀. Since 𝑀 is scalar, then there exist 𝑎 ∈ 𝑅 such that 𝑓 𝑚 𝑚𝑎 for all 𝑚 ∈ 𝑀. Then 𝑓 𝑀 𝑀𝑎 implies 𝑀𝑎 ⊆ 𝐾 and hence 𝑓 𝑀 ⊆ 𝐾 as desired. Theorem (5.7): Let 0 𝑀 be an 𝑅-module such that 𝑆 is commutative. If 𝑀 is a regular 𝑆- module then 𝑀 is 𝑆-semisecond. Proof. Similarly proof of Theorem 4.1. Corollary (5.8): Every Rickart and dual Rickart module has a commutative endomorphism ring is 𝑆-semisecond.   92  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (2) 2020 Proof. By [16]. The endomorphism ring of Rickart and dual Rickart modules is von Neumann regular so by Theorem 5.7, the result is obtained. Remark (5.9): The commutativity condition in Theorem 5.7 or Corollary 5.8 can not (cannot) (be) dropped as follows, 𝑀 ℤ ⊕ ℤ as ℤ-module is Rickart and dual Rickart and hence 𝑆 𝐸𝑛𝑑ℤ ℤ ⊕ ℤ 𝐸𝑛𝑑ℤ ℤ 𝐻𝑜𝑚ℤ ℤ , ℤ 𝐻𝑜𝑚ℤ ℤ , ℤ 𝐸𝑛𝑑ℤ ℤ ≅ 𝑀𝑎𝑡 ℤ ℤ ℤ ℤ ℤ is von Neumann regular, but 𝑀𝑎𝑡 ℤ not commutative ring. On the other hand, ℤ ⊕ ℤ ≅ , , , , so if we take 𝑓 0 1 0 0 ∈ 𝑆 𝐸𝑛𝑑ℤ ℤ ⊕ ℤ implies 𝑓 𝑀 0 1 0 0 , 𝑥, 𝑦 ∈ ℤ , 𝑓 𝑀 0 0 0 0 , 𝑥, 𝑦 ∈ ℤ it follows that 𝑀 is not semisecond as 𝑆-module that is, 𝑀 is not 𝑆-semisecond as ℤ-module. Proposition (5.10): Every non-zero direct summand of 𝑆-semisecond module is 𝑆- semisecond. Proof. Let 𝑁 be a direct summand of an 𝑆-semisecond 𝑅-module 𝑀 then 𝑀 𝑁 ⊕ 𝐻 for some submodule 𝐻 of 𝑀. Let 𝑓 ∈ 𝐸𝑛𝑑 𝑁 with 𝑓 𝑁 ⊆ 𝐾 for some 𝐾 a submodule of 𝑁. We can define 𝛼 𝑛 ℎ 𝑓 𝑛 where 𝑛 ∈ 𝑁 and ℎ ∈ 𝐻. It is easy to see that 𝛼 ∈ 𝑆, 𝛼 𝑀 𝑓 𝑁 implies 𝛼 𝑀 𝑓 𝑁 ⊆ 𝐾. It follows 𝛼 𝑀 ⊆ 𝐾 implies 𝛼 𝑀 ⊆ 𝐾 and hence 𝑓 𝑁 ⊆ 𝐾 as desired. Theorem (5.11): The following statements are equivalent (1) 𝑀 is a 𝑆-semisecond 𝑅-module. (2) 𝑀 0 and 𝐾: 𝑀 is a semiprime ideal of 𝑆 for each proper submodule 𝐾 of 𝑀. Proof. Similarly, proof of Theorem 3.1. Corollary (5.12): If 𝑀 is an 𝑆-semisecond 𝑅-module 𝑀 then 𝑎𝑛𝑛 𝑀 𝑓 ∈ 𝑆: 𝑓 𝑀 0 is a semiprime ideal of 𝑆. Proof. Directly By Theorem 5.11. Examples (5.13): The opposite result is not held in general for example ℤ is not semisecond and hence not 𝑆-semisecond while 𝑎𝑛𝑛 ℤ 0 is a semiprime ideal of 𝑆. Corollary (5.14): If 𝑀 is an 𝑆-semisecond 𝑅-module then for every proper submodule 𝐾 of 𝑀 we have 𝐾: 𝑀 𝐾: 𝑔 𝑀 for each 𝑔 ∈ 𝑆. Proof. Similarly, proof of Corollary 3.5. Corollary (5.15): If 𝑀 is an 𝑆-semisecond 𝑅-module then 𝑎𝑛𝑛 𝑀 𝑎𝑛𝑛 𝑔𝑀 for each 𝑔 ∈ 𝑆. Proof. Directly by Corollary 5.14. Theorem (5.16): The following statements are equivalent (1) 𝑀 is an 𝑆-semisecond 𝑅-module. (2) 𝑀 0 and for each ideals 𝐼 of 𝑆 and 𝐾 a submodule of 𝑀 such that 𝐼 𝑀 ⊆ 𝐾 implies 𝐼𝑀 ⊆ 𝐾. Proof. Similarly, proof of Theorem 3.7. Corollary (5.17): The following statements are equivalent (1) 𝑀 is an 𝑆-semisecond 𝑅-module. (2) 𝑀 0 and for each ideals 𝐼 of 𝑆 and 𝐾 a proper submodule of 𝑀 and 𝐼 ⊆ 𝐾: 𝑀 implies 𝐼 ⊆ 𝐾: 𝑀 . Proof. Directly via Theorem 5.16. Proposition (5.18): The following statements are equivalent   93  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (2) 2020 (1) 𝑀 is an 𝑆-semisecond 𝑅-module. (2) 𝑀 0 and for each ideal 𝐼 of 𝑆 implies 𝐼 𝑀 𝐼𝑀. Proof. By using Theorem 5.10 and Theorem 5.2. 6. Conclusion In this research we present comprehensive study of semisecond submodules. 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