On Semisecond Submodules

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


Introduction
is indicated a ring with identity and is viewed as a non-zero --bimodule where the endomorphism ring of . We use the notation ʻʻ ⊆ ʼʼ to denote inclusion. A non-zero submodule of is said to be a second submodule if for any ∈ , the endomorphism : → defined by for each ∈ , is either surjective or zero (that is or 0) [1]. Equivalently 0 is a second submodule of if or 0 for every ideal of [1]. In that situation, is a prime ideal of [1]. A non-zero module is second (or coprime) if 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 of is weakly second submodule whenever ⊆ where , ∈ and a submodule of implies either ⊆ or ⊆ [2]. Equivalently, a non-zero submodule of is called weakly second if or for every , ∈ [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 of is wekly prime whenever ⊆ where , ∈ and a submodule of implies either ⊆ or ⊆ [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 of is weakly secondary submodule if ⊆ where , ∈ and is a submodule of implies either ⊆ or ⊆ for some positive integer . A nonzero submodule is a secondary submodule of if for any ∈ , the endomorphism : → defined by for each ∈ , is either surjective or nilpotent ( that is or 0 for some positive integer ) [1]. Equivalently, 0 is a secondary submodule of if for every ideal of , or 0 for some positive integer [1]. In this case, is a primary ideal of (that is is a prime ideal of ) [1]. A proper submodule of is classical primary if ⊆ where , ∈ and is a submodule of then ⊆ or ⊆ for some positive integer [6]. A proper submodule of is called completely irreducible when ⋂ ∈∧ where ∈∧ is a family of submodules of implies that for some ∈∧ [2]. It is not hard to see that every submodule is an intersection of completely irreducible submodules of consequently the intersection of all completely irreducible submodules of is zero. is called simple (sometimes minimal) submodule of a module if 0 and for each submodule of and contains properly implies 0 [7]. is coquasi-dedekind if all nonzero endomorphism of is epimorphism (in other word, for every 0 ∈ ) [8]. Let be a commutative integral domain, is called divisible module over if for each 0 ∈ [7]. A proper submodule is maximal if it is not properly contained in any proper submodule of [7]. A proper submodule is called prime if ∈ implies ∈ or ⊆ [9]. is called a prime module if the zero submodule is prime. A proper ideal is prime if ∈ where , ∈ implies ∈ or ∈ [10]. Equivalently, a proper ideal is prime if ⊆ where and are ideals of implies ⊆ or ⊆ [10]. A ring in which every ideal prime is called fully prime [11]. Equivalently, a ring is fully prime if and only if it is fully idempotent (a ring in which every ideal is an idempotent that is for each ideal of ) and the set of ideals of is totally ordered under inclusion [11]. A proper submodule is called primary if ∈ implies ∈ or ⊆ for some positive integer [6].
is called a primary module if the zero submodule is primary. A proper ideal is primary if ∈ where , ∈ implies ∈ or ∈ for some positive integer [6]. 0 is called an -second module if for every ∈ implies or 0 [12]. 0 is called an S-weakly second module whenever ⊆ , where , ∈ and a submodule of implies either ⊆ or ⊆ [3]. Equivalently, is an S-weakly second module if and only if for each , ∈ implies or ⊇ [3]. is called multiplication when each submodule of , we have for some ideal of [13]. We able to take : ∈ and ⊆ is an ideal of [13]. is called faithful if 0: ∈ and 0 0. is a scalar module when for each ∈ there is ∈ with for all ∈ [14].
The aim of this research is to continue studying the concept of semisecond submodules. A nonzero submodule of is called semisecond if for each ∈ , [2]. A nonzero module is said to be semisecond if is semisecond submodule of itself. In fact this idea is the dual notion of the concept semiprime submodules. A proper submodule of is called semiprime if for each ∈ , ∈ such that ∈ implies ∈ [9]. A proper ideal of is semiprime if for each ∈ such that ∈ implies ∈ [7]. Equivalently, a proper ideal of is semiprime if for each ideal of such that ⊆ implies ⊆ [7]. It is well-known that is fully semiprime (that is in which every ideal is semiprime ) if and only if is von Neumann regular ( that is for every ∈ , there is ∈ such that ) [15]. It is well-known if is commutative then is von Neumann regular if and only if if and only if every ideal of is pure ( that is ∩ for each ideal and of ) if and only if is fully idempotent. And is called regular if for every ∈ and for every ∈ we have for some ∈ . If is regular then every submodule of is pure (that is every submodule of satisfying ∩ for each ideal of ) [15]. If is commutative then is regular if and only if for every ∈ and for every ∈ we have for some ∈ . Also is Boolean ring if for every ∈ [7]. Thus a Boolean ring is von Neumann. We call a module is Rickart when for every ∈ , is a direct summand of [16]. is a dual Rickart module when for every ∈ , Im is a direct summand of [16]. It is wellknown that for each ∈ we can define : → by for each ∈ then . This means is von Neumann regular if and only if is dual Rickart asmodule. A nonzero submodule of is weak semisecond whenever ⊆ where ∈ and a submodule of implies either ⊆ or ∈ [17]. A nonzero submodule of is called a strongly 2-absorbing second submodule if for each , ∈ , we have or or 0 [18]. A module is called cacellation if implies for each ideal and of [19]. Other works within [20][21][22][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 we denote respectively, integers, rational numbers, the -Prüfer group, the residue ring modulo and an matrix ring over .

Semisecond Submodules
We give a characterization of semisecond submodules, first we recall the main definition.

Theorem (2.2):
The following assertions are equivalent (1) is a semisecond submodule of an -module (2) 0 and whenever ⊆ , where ∈ and a submodule of implies ⊆ Proof.

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 . ℤ .
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, ℤ ⊕ ℤ as ℤ-module is semisecond but it is not strongly 2absorbing second,(and hence not weakly second ) since . 3 2.3 0 ⊕ ℤ . 2 and . 2.3 0 . (8) If is a maximal (and hence prime ) submodule then may not be semisecond. For example, ℤ is a maximal submodule of ℤ as ℤ-module but is not semisecond since for every ∈ ℤ and any prime number . (9) Let and be submodules of an -module with ⊆ ⊆ . If is a smisecond submodule of then needs not be a semisecond submodule of . Let ℤ . 2 and ℤ submodules of ℤ as ℤ-module where is a simple submodule so it is semisecond while is not semisecond by (5). (10) Let and be submodules of an -module with ⊆ ⊆ . If is a sermisecond submodule of , then needs not be a semisecond submodule of . Let ℤ be a submodule of ℤ as ℤ-module. Since is a divisible module then is semisecond but is not semisecond because . 0 ℤ . (11) As another example of (10), ℚ as ℤ-module is divisible so it is semisecond but the submodule ℤ is not semisecond.

Theorem (2.4):
The following assertions are equiavalent (1) is a semisecond submodule of an -module . (2) 0 and for each , ∈ and a finite intersection of completely irreducible submodules of with ⊆ implies ⊆ .  (1) is a semisecond submodule of -module . (2) is a semisecond submodule of -module for each submodule of contained in .
Proof. (1)  (2) Let be a semisecond submodule and : → be the natural homomorphism for each submodule of contained in so by Proposition 2.5, is a semisecond submodule .

More Characterizations and Facts About Semisecond Submodules Theorem (3.1):
The following statements are equivalent (1) is a semisecond submodule of an -module . (2) 0 and : is a semiprime ideal of for each submodule ⊉ in . Proof.
(1)  (2) Assume is a semisecond submodule of an -module and a submodule of such that ⊈ implies : . Let ∈ with ∈ : implies ⊆ thus ⊆ and hence ∈ : as required.
(2)  (1) Let and be submodules of an -module such that ⊆ where ∈ . In case ⊆ then already ⊆ . If ⊈ then : is a semiprime ideal of by hypothesis and ∈ : implies ⊆ as desired. Corollary (3.2): Every submodule of a module over a fully semiprime (that is von Neumann regular) ring is semisecond.

Corollary (3.3):
If is a semisecond submodule of an -module then is a semiprime ideal of . Proof. Directly via Theorem 3.1.

Examples (3.4):
0 is a semiprime ideal of ℤ for every nonzero submodule of the ℤ-module ℤ while is not semisecond.

Corollary (3.5):
If is a semisecond submodule of an -module then for every submodule ⊉ in we have : : then ⊆ implies for each ∈ ⊆ so ∈ : . Conversly, let ∈ : then ⊆ implies ∈ : and we can take then ∈ : . Via Theorem 3.1, : is a semiprime ideal of implies ∈ : as required.

Corollary (3.6):
If is a semisecond submodule of an -module then for each ∈ . Proof. Directly by Corollary 3.5. Theorem (3.7): The following statements are equivalent (1) is a semisecond submodule of an -module . (2) 0 and for each ideals of such that ⊆ implies ⊆ . Proof. (1)  (2) First since is a semisecond submodule of an -module then 0. Let be an ideal of and a submodule of . If ⊈ we have either ⊈ and so nothing to prove or ⊆ it follows ⊆ : and by Theorem 3.1, : is a semiprime ideal of so ⊆ : and hence ⊆ . In case ⊆ then the result already is obtained.
(2)  (1) Let ⊆ , where ∈ and a submodule of , then ⊆ . By hypothesis ⊆ where is the principal ideal generated by and hence ⊆ as dsired.

Corollary (3.8):
The following statements are equivalent (1) is a semisecond submodule of an -module . (2) 0 and for each ideal of and a submodule of such that ⊈ and ⊆ : implies ⊆ : . Proof. Directly via corollary 3.7. Corollary (3.9): The following statements are equivalent (1) is a semisecond submodule of an -module . (2) 0 and for each ideal of implies .
Proof. (1)  (2) First since is a semisecond submodule of an -module then 0. Let be an ideal of then ⊆ so by Theorem 3.7, we have ⊆ and thus .
(2)  (1) it is clear. Theorem (3.10): Let be a submodule of an -module . If for each ∈ , then is semisecond. Proof. Assume for each ∈ , then for some , ∈ implies ∈ and hence . Theorem (3.11): If is a semisecond finitely generated submodule of an -module then for each ∈ , . Proof. Let ∈ then that is . By hypothesis is finitely generated. It is not hard to see that is also finitely generated. Via [23, Corollary 2.5], it follows that 1 ∈ and 0. Let 1 for some ∈ then 1 implies 1 0. This means ∈ so for some ∈ implies and hence ⊆ . Then .

Theorem (3.12):
Let be a finitely generated submodule of a module over a commutative ring . The following statements are equivalent (1) is semisecond. (2) For each ∈ , for some ∈ . Proof. (1)  (2) By Theorem 3.11, for each ∈ , . Then for some , ∈ and , ∈ . By choosing 1 we have for some ∈ thus ⊆ implies ⊆ hence . Put it follows ∈ . Therefore but that is ∈ thus as desired.

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 be a nonzero regular -module. We show for each ∈ . Let ∈ implies for some ∈ it follows ∈ for some ∈ .

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 (3) Semisecond modules may not be semisimple. Consider ∏ ∈∧ is commutative von Neumann regular ring ( is a regular as -module ) and hence is semisecond but is not semisimple since the submodule ⊕ ∈∧ is not a direct summand of . Proposition (4.9): Let be a commutative ring then we have the equivalent (1) is von Neumann regular. (2) is fully semiprime.

Example (4.11): Let
is not a pure submodule of .

Corollary (4.12):
Let be an -module and be an ideal of such ⊆ . If is a semisecond ring then is semisecond.
Proof. Since is considered as -module so by Proposition 4.10, the result is obtained.

Proposition (4.13):
Let be an -module and be an ideal of such that ⊆ . Then is a semisecond -module if and only if is a semisecond -module.
(2) Consider ℤ as ℤ-module implies ℤ ℤ ℤ ℤ is semisecond but ℤ is not semisecond. Proposition (4.15): If is a cancellation semisecond submodule of an -module then is semisecond. Proof. For each ∈ , we have , then and since is cancellation implies as desired.

Corollary (4.16):
If is a finitely generated faithful multiplication semisecond -module then is fully idempotent (and hence semisecond). Proof. Let be a semisecond -module then for each ideal of . Since is a finitely generated faithful multiplication so by [13], is cancellation then thus is fully idempotent.

Corollary (4.17):
If is a cancellation (or finitely generated faithful multiplication) semisecond -module such that the set of ideals of is totally ordered under inclusion then is fully prime. Proof. By Corollary 4.15, is fully idempotent so by [11]. is fully prime. Theorem (4.18): Let be a multiplication -module. If : is a semisecond ideal of then is a semisecond submodule of . Proof. By hypothesis, : : for each ideal of then : : . By hypothesis is multiplication thus so is semisecond. Theorem (4.19): Let be a finitely generated faithful multiplication -module. If is a semisecond submodule of then : is a semisecond ideal of . Proof. Since for each ideal of then : : because is multiplication. But is finitely generated faithful implies is cancellation and hence : : thus : is semisecond.

Remark (4.20):
If is a semisecond ideal of then . Proof. Since for each ideal of so if we choose implies . Proposition (4.21): Every nonzero pure submodule of a semisecond module is semisecond.
Proof. Let be a nonzero pure submodule of a semisecond -module . Then for each ideal of implies ∩ ∩ as desired. The following result is appeared in [2]. Without proof Proposition (4.22): Every sum of second submodules is semisecond. Proof. Let ∈ , and be second submodules of an -module implies either or 0 ⊆ or 0 0 0 ⊆ and hence . 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 ∈ and be a semisecond submodule of a prime -module implies then for each ∈ we have for some ∈ implies 0. But 0 is a prime submodule in, it follows either ∈ 0 implies and hence or ∈ 0 : ⊆ 0 : implies 0 as desired. Proposition (4.25): Every semisecond submodule of primary module is secondary. Proof. Similarly of Proposition 4.24.
(2)  (1) Assume ⊆ , where ∈ and a submodule of implies ⊆ as required. Proposition (5.3): Every semisecond multiplication module is -semisecond. Proof. Let be a semisecond multiplication -module and ∈ with ⊆ for some a submodule of . Since is multiplication then for some ideal of and hence ⊆ . By Theorem 3.6, we have ⊆ it follows ⊆ that is is -semisecond. Corollary (5.4): Every semisecond cyclic module is -semisecond.

Corollary (5.12):
If is an -semisecond -module then ∈ : 0 is a semiprime ideal of . 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 -semisecond while ℤ 0 is a semiprime ideal of . Corollary (5.14): If is an -semisecond -module then for every proper submodule of we have : : for each ∈ . Proof. Similarly, proof of Corollary 3.5.

Conclusion
In this research we present comprehensive study of semisecond submodules. We show that every regular module is semisecond, and the semisecond and regular concepts in the commutative rings are the same. Comprehensive study in this type of modules is introduced and numerous examples and basic properties are provided.