On Weakly Second Submodules

    Let  be a non-zero right module over a ring  with identity. The weakly second submodules is studied in this paper. A non-zero submodule  of   is weakly second Submodule when  ,  where ,  and  is a submodule of  implies either  or   . Some connections between these modules and other related modules are investigated and number of conclusions  and characterizations are gained.


Introduction
is denoted a ring has an identity and is studied as a non-zero left -right -bimodule where the endomorphism ring of . We use the notation to denote inclusion. is said to be a second submodule of if for any , the endomorphism defined by for each , is either surjective or zero ( that is or ) [1]. Equivalently is a second submodule of if or for every ideal of [1]. In that situation, is a prime ideal of [1]. A non-zero module is a second (or coprime ) if is a second submodule of itself [1]. As a new type of second submodules, the concept of weakly second submodules was presented and studied in [2]. A non-zero submodule of is weakly second submodule whenever where , and a submodule of implies either or [2]. A non-zero module is a weakly second module if is a weakly second submodule of itself [2]. In fact this idea as a dual notion of the concept weakly prime ( sometimes is called classical prime ) submodules. A proper submodule of is wekly prime whenever where , and a submodule of implies either or [3]. In [4], we provide the idea of weakly secondary as a generalization of weakly second concept and in the same time it is a new type of secondary submodules and a dual notion of classical primary submodules respectively. A nonzero ISSN: 0067-2904

Saadi and Ahmed
Iraqi Journal of Science, 2019, Vol. 60, No.8, pp: 1791-1801 28:3 submodule of is weakly secondary submodule whenever where , and is a submodule of implies either or for some positive integer [4]. is a secondary submodule of if for any , the endomorphism defined by for each , is either surjective or nilpotent ( that is or for some positive integer ) [1]. Equivalently, is secondary of if for every ideal of , or for some positive integer [1]. A proper submodule of is classical primary whenever where , and is a submodule of then or for some positive integer [5]. is called simple ( sometimes minimal ) submodule of a module if and for each submodule of and contains properly implies [6]. A module is called simple module if is simple submodule of itself [6]. is coquasi-dedekind if all non-zero endomorphism of is epimorphism (in other word, for every ) [7]. Let be a commutative integral domains, is called divisible module over if for each [6]. A proper submodule is maximal if it is not properly contained in any proper submodule of [6]. A proper submodule is called prime if implies or [8]. A proper ideal is prime if where , implies or [9]. Equivalently, a proper ideal is prime if where and are ideals of implies or [9]. A ring in which every ideal is prime is called fully prime [10]. Equivalently, a ring is fully prime if and only if it is fully idempotent and the set of ideals of is totally ordered under inclusion [10]. is comultiplication provided that for each submodule of , there exists an ideal of such that [ ] and is a submodule of [11]. We able to take [ ] and is an ideal of [11]. is called a submodule pure in an -module when for each ideal of [12]. is called regular when every submodule of is pure [12]. is called -second if every implies or [13]. is indecomposable if and it cannot be written as a direct sum of non-zero submodules ( that is 0 and are the only direct summands ) [6]. is called multiplication when each submodule of , we have for an ideal of [14]. We able to take [ ] and [14]. is a scalar module when for each there is with for all [15]. Other studies within [16][17][18][19][20][21][22][23][24][25][26] is related topics.
The paper consists of five parts. Within part two, we investigate the weakly second submodules idea and we supply examples (Remarks and Examples 2.3) and needful features of this concept. We add a new characterization (Proposition 2.9) and some properties of this concept (Proposition 2.4). The direct sum of weakly second submodules is discussed (Proposition 2.5). In Section three more characterizations is given ( Theorem 3.1, Theorem 3.7 and Theorem 3.8 ). In section four we look for any relationships between weakly second submodules and related modules such as (Proposition 4.1 and Proposition 4.4). S-weakly second modules is dfined and basic properties about this modules is studied in section five. In what follows, ℤ, ℚ, ℤ , ℤ ℤ ℤ and we denote respectively, integers, rational numbers, the -Prüfer group, the residue ring modulo and an matrix ring over .

Weakly Second Submodules
Main facts of this part are introduced. We begin by the following. Definition 2.1 [2] A nonzero submodule of is a weakly second submodule whenever , where , and is a submodule of implies either or . Theorem 2.2 [2] The following statements are equivalent (1) is a weakly second submodule of . (2) and for each , implies or . Remarks and Examples 2.3 (1) (2) Weakly second submodules fail to be second. Consider ℤ ℤ as ℤ-module where is a prime number then is weakly second since or for each , ℤ but is not second since if then ℤ (3) As another example of (2), the submodule ℤ ℤ is weakly second of ℤ ℤ as ℤ-module but is not second. (4) Clearly every weakly second submodule is weakly secondary while the converse is not true by [3]. (5) Clearly weakly second and weakly secondary concepts are coincide over Boolean rings. (6) The secondary submodules and weakly second concepts do not imply from each one to another. The ℤ-module ℤ is secondary since ℤ ℤ or ℤ for some a positive integer but ℤ is not weakly second because ℤ while ℤ . On the other side, ℤ ℤ as ℤmodule is weakly second but not secondary. Since for each , ℤ, if and are not multiple of implies  for each positive integer but when or is a multiple of , we have ℤ  and for each positive integer . (7) The following implication is clear simple submodule  second submodule  weakly second. (8) The following implication is clear coquasi-dedekind module  divisible module  second module  weakly second module. (9) It is clear ℤ and ℚ as ℤ-modules are coquasi-dedekind ( and hence are divisible ) by (8) they are weakly second. Further it is well known that every direct summand of divisible module is divisible [6]. And every product ( or sum ) of divisible modules is divisible [6]. Accordingly, ℤ ℤ (where and prime numbers) and ℚ ℚ as ℤ-modules are divisible and hence weakly second. (10) If is weakly second module then need not be coquasi-dedekind. For example ℚ ℤ ∑ ℤ as ℤ-module is divisible and hence it is weakly second but it is not coquasi-dedekind. (11) If is a maximal (and hence prime) submodule then may not be weakly secondary. For example, ℤ is a maximal submodule in ℤ as ℤ-module but is not weakly second since and neither nor . (12) Let and be submodules of an -module with . If is weakly second then need not be weakly second. For example, let ̅ ̅ ̅ and ℤ submodules of ℤ as ℤ-module where is a simple submodule so it is weakly second while is not weakly second because and and ̅ ̅ . (13) Let and be submodules of an -module with . If is weakly second then need not be weakly second submodule of . For example, let ℤ ℤ and ℤ ℤ be submodules of ℤ ℤ as ℤ-module where and prime numbers. Since is a divisible module then is weakly second but is not weakly second because while ℤ and ℤ . (14) As another example of (13), ℚ as ℤ-module is divisible so it is weakly second but the submodule ℤ is not weakly second. Proposition 2.4 Every nonzero homomorphic image of weakly second submodule is weakly second.

Proof
Let and be -modules and an -homomorphism. Let be a weakly second of . (1) The direct sum of weakly second submodules need not be weakly second. For example, ℤ and ℤ as ℤ-modules are weakly second where and are prime numbers then ℤ ℤ is not weakly (2) In general ℤ ℤ as ℤ-module is not weakly second for each positive integers .
(3) Obviously, if is a square-free integer (an integer which has a prime factorization has exactly one factor for each prime that appears in it) then ℤ as ℤ-module is not weakly second. Oppositely fails in general, ℤ as ℤ-module is not weakly second because ℤ but ℤ ℤ and is not square-free.
Let be a direct sum of two -modules and . If is a weakly second submodule of then may be not a weakly second submodule of . For example ℚ is a divisible ℤ-module so it is weakly second while ℚ ℤ is not a weakly second since [ℚ ℤ ℤ ℚ ℤ] ℤ is not a prime ideal of ℤ then by Theorem, ℚ ℤ is not a weakly second ℤ-module. In fact for any -module then ℤ is not a weakly second ℤ-module. (5) Let be a direct sum of two -modules and . If is divisible ( or weakly second ) of and is not weakly second of then is not weakly second of .

Proof
Suppose is a weakly second submodule of then for each , we have or . It follows or which is a contradiction because is not a weakly second submodule of as desired. (6) ℚ ℤ, ℚ ℤ , ℤ ℤ and ℤ ℤ as ℤ-modules are not weakly second by (4) where is a square-free integer. Proposition 2.8 If is a weakly second submodule of then is a weakly second submodule of as -module.

Proof
Firstly because . Let , then but is weakly second implies either or and hence or as required. Proposition 2.9 The next are equivalent (1) is a weakly second submodule of .
(2) is a weakly second submodule of for each submodule of contained in . Proof (1)  (2) Let be a weakly second submodule and be the natural homomorphism for each submodule of contained in so by Proposition, is a weakly second submodule .
(2)  (1) It is clear by taking . The following statements are equialent (1) is a weakly second submodule of an -module . (2) and for each ideals and of implies or .

Proof
Similarly to the proof of Theorem 2.2 and by Theorem 3.1. Corollary 3. 10 The following statements are equialent (1) is a weakly second of an -module . (2) and

Weakly Second Submodules and Related Concepts
The following result is given in [11], we give the details of the proof. Proposition 4.1 If is a non-zero comuliplication submodule of together with is prime of then is second.

Proof
Let . For every we can define the endomorphism by for each then . Because is comultiplication implies for an ideal of so follows . But is prime so or . In case then follows as desired. Corollary 4.2 Let be a comuliplication -module such that the annihilator of any non-zero submodule of is a prime ideal of then every nonzero submodule is second.

Proof
Because every submodule of a comultiplication module is comultiplication then by Proposition 4.1, the result is obtained. Let be a non-zero pure submodule of a weakly second -module . Then for each ideals and of implies or . It follows either or as desired. Corollary 4.5 Each submodule of a regular weakly second module is weakly second. Corollary 4.6 Any submodule of a semisimple weakly second module is weakly second. Example 4.7 ℤ as ℤ-module is semisimple but not weakly second as shown in Remark and Example 2.3 (12) confirms that the status weakly second in Corollary 4.6 can not omitted.

-Weakly Second Modules
At this point we define S-weakly second modules. Firstly we supply a characterization and examples of -second modules. Theorem 5.1 The following are equivalent (1) is an -second module. (2) and whenever where and a submodule of implies either or .  (2) is not valid in general. ℤ ℤ as ℤ-module is second but not S-second because there is an endomorphism (3) Every S-Second module is indecomposable ( that is when a module has a decomposition then is not -second ).

Proof
Let be an S-second -module then . Suppose that for some -modules and . So we can define the map maps by then implies and hence is not S-second which is a contradiction. (4) The counter of (3) is not correct comprehensively. ℤ and ℤ as ℤ-modules are indecomposable but not second and hence it is not S-second. (5) Evidently coquasi-dedekind module is S-second.  (5) is not correct generally. Let be a field and let be the set of infinite matrices over that have the form ( ) Where is any finite matrix and is any element of . It is not hard to see that is a ring with identity and the only non-zero proper ideal of is the subset of all matrices of of the form ( ) so is clear and hence is prime [10], also it is obvious the zero ideal is prime and hence is fully prime ring. Via Theorem 3.1, is a weakly second which is not second. (7) We have the implication coquasi-dedekind modules  -second modules  -weakly second modules  indecomposable modules. Theorem 5.5 Study the equivalent (1) is an -weakly second -module. (2) and for each , implies or .

Proof
(1)  (2) Assume is an -weakly second -module then . Let , and for submodule of . We can choose so by (1) or and hence or .
(2)  (1) Let and , with for submodule of . By (2), or as desired. Corollary 5. 6 If is commutative ring we have the equivalent (1) is an -weakly second -module. (2) and for each , implies or . Proof It is obvious Theorem 5.7 The following statements are equivalent (1) is an -weakly second -module.. (2) and [ ] is a prime ideal of for each proper submodule of . (1) The opposite of corollary 5.8 is not hold in general.
ℤ is a prime ideal of ℤ ℤ ℤ which is not weakly second and hence it is not -weakly second .