On Dense Subsemimodules and Prime Semimodules

In this paper, we study the class of prime semimodules and the related concepts, such as the class of semimodules, the class of Dedekind semidomains, the class of prime semimodules which is invariant subsemimodules of its injective hull, and the compressible semimodules. In order to make the work as complete as possible, we stated, and sometimes proved, some known results related to the above concepts.


Introduction
Throughout this paper, will denote a commutative semiring with identity, and is ansemimodule.
This paper consists of three sections. In Section one, we introduce some definitions and remarks which we will use in the paper. In Section two, we introduce the concept of density of semimodules. A non-zero -subsemimodule of an -semimodule is said to be dense in , if ∑ , where the sum is taken over all . We use the density concept to define the class of semimodules, as is said to be semimodule if each non-zero subtractive subsemimodule of is dense in . In Section three, we define the concept of prime semimodules, analogous to that in modules [4], where is said to be prime if ann ann( ), for each non-zero subtractive subsemimodule of . Similar to that in modules [1], we will show that every semimodule is a prime semimodule. The aim of this paper is to discuss the converse of this statement in the case of semimodules having injective hull. Also we generalize some types of prime modules for semimodules, such as the compressible type.

ISSN: 0067-2904
(by the choice of and ), hence is not subtractive. On the other hand, it is clear that any cyclic ideal of is subtractive. Remark 1.11. Let be a subsemimodule of the -semimodule , and let be the smallest non-zero element of , then either or . Proof: Assume that , then , if is the smallest element greater than such that , , then . Similarly proceeding, we have . Remark 1.12.
Let be a commutative semiring with identity. A set S is said to be a multiplicatively closed set of provided that "if , then . The localization of at (R S ) is defined in the following way: First define the equivalent relation on by , if for some . Then put R S as the set of all equivalence classes of and define the addition and multiplication on R S , respectively, by and , where is also denoted by , by which we mean the equivalence class of . It is, then, easy to see that R S with the above mentioned operations of addition and multiplication on R S is a semiring [15]. Definition 1.13. In Remark 1.12, if is the set of all not zero divisors of , then the total quotient semiring of the semiring is defined as the localization of at . Note that is also ansemimodule. For more details, see previous articles [11,13]. Definition 1.14. A subset of the total quotient semiring of is called fractional ideal of a semiring , if the following hold: 1. is an -subsemimodule of , that is, if and , then and . 2. There exists a non-zero divisor element such that . Let , be two fractional ideals of a semiring . Then . It is clear that any ideal of is a fractional ideal of a semiring . Definition 1.15. Let be a fractional ideal of a semiring , then is called invertible if there exists a fractional ideal of such that . Note that is unique and we denote that by . For more details, see for example earlier works [10,11].

Semimodules
Let be a family of -semimodules. Birge Zimmermann-Huisgen [3] introduced the definition of self-generator formodules. In this paper, we recall this definition for -semimodules. is called a self-generator if generates each of its subsemimodules. In other words, an -semimodule is called self-generator, if for any subsemimodule of , ∑ . In this section, we study the semimodules which can be generated by each of their non-zero subsemimodules. This is a "dual proplem" of self-generator concept.  (1), assume that and is any integer with , then , (which is not possible). Hence must equal zero. Therefore . Using the same way we prove(2). The following example shows that the condition in Proposition 2.4 is not sufficient. Example 2.6. Let be considered as a -semimodule, where and are the groups of integers and rationals, respectively. Let N=0+ be a non-zero subsemimodule of . It is clear that, ann ann . If , . From Remark 2.5, we have , then . Thus is not dense in . Note that, in Example 2.6, if we put considered as a -semimodule, where is a semigroup of natural numbers, we will get is not dense in . The following lemma shows that the condition of Proposition 2.4 is sufficient to make a subsemimodule dense if a subsemimodule is cyclic. Lemma 2.7. Let be a non-zero cyclic subsemimodule of an -semimodule , then the following statements are equivalent: 1.
The assumption implies that is well-defined. Finally, suppose that (3) holds , then it is clear that , let by (3), then for all , we have ∑ , thus . Thus . After defining the concept of a dense subsemimodule, as previously described in the modules [1, page 11], we are ready now to give the concept of a semimodule, which is a dual, in some sense, to the concept of self-generator semimodule, given in modules. Definition 2.8. An -semimodule is said to be a semimodule if for each non-zero subtractive subsemimodule of , , i.e. each non-zero subtractive subsemimodule of is dense in . Note that is a semimodule if it is generated by each of its nonzero subtractive subsemimodule, while is a self-generator if it generates each of its subtractive subsemimodules. be the set of rationals of the form , with and are in and is not divisible by . Then is a subsemigroup of . As a -module [17]. We put . Then is a -semimodule. It is known that each proper non-zero subsemigroup of is cyclic of the form [17]. Note that, since each element of where is of order less than or equal to , then is not dense in . Thus is not a semimodule. A subsemimodule of an -semimodule is called invariant subsemimodule if , , and is called a stable subsemimodule if , [2]. Remark 2.10. Let be a non-zero subsemimodule of an -semimodule , then 1.
is a stable subsemimodule of . Proof: (1) and (2)  The following proposition relates the concept of a semimodule and the concept of stability. Proposition 2.11. Let be an -semimodule, then is a semimodule iff has no non-trivial stable subsemimodules. Proof: Assume that is a semimodule, and is a proper non-zero stable subsemimodule of . By Remark 2.10, . Since is semimodule, hence , which is a contradiction. Conversely, since is a stable non-zero subsemimodule of , see Remark 2.10, thus by assumption, . Therefore is semimodule. Now, we study when an ideal is dense in semiring. Remark 2.12. A non-zero ideal of a semiring is dense in iff trace . Golan [9, page 39] proved thatan ideal of a ring is a direct summand iff for some idempotent element of . Here, we use another proof for a semirings. Lemma 2.13. An ideal of is a direct summand iff for some idempotent element of . Proof: ( ) Assume that is a direct summand of , that is , then ́ for some . Therefore. is a direct summand of . As in the modules, we give the following lemma without proof, since it is already included in the modules [9, page 61]. Lemma 2.14. A left -semimodule is isomorphic to a direct summand of a free left -semimodule iff it is projective. Theorem 2.15. Let be a non-zero subtractive ideal of , then is dense in iff is a faithful finitely generated projective ideal.

Proof:
Suppose that is dense in , by Remark 2.12, ∑ , where , , for finite . Thus, ∑ ∑ . Hence is finitely generated, and by the dual basis lemma, is projective, [5]. Since ann ann , thus is faithful. ( ) Suppose that is a faithful finitely generated projective ideal. Since faithful, then ann . Since is projective, then by Lemma 2.14, we have is a direct summand of . . Hence, is an ideal of , and . Therefore, is invertible ideal of . The following two corollaries are immediate from Remark 2.19 and Proposition 2.17. Corollary 2.20. Every principal ideal in a semiring generated by a non-zero divisor is dense in . Corollary 2.21. Let be a semiring, then the following statements are equivalent: 1.
is a semidomain. 2. Each non-zero principal ideal of is an invertible ideal of . 3. Each non-zero principal ideal of is dense in .

Prime Semimodules Having Injective Hull
In Proposition 2.4, we saw that for every dense subsemimodule of , ann ann( ), thus in a -semimodule , for every non-zero subtractive subsemimodule of , ann ann( ). And in Lemma 2.7, we observed that a cyclic subsemimodule is dense in iff ann ann(M). These observations lead us to study prime semimodules. Analogous to the concept of prime modules [4], we define a prime semimodules as follows: Definition 3.1. An -semimodule is said to be prime semimodule if ann ann , for every non-zero subtractive subsemimodule of .
We observed that the class of prime semimodules contains the class of semimodules. But the converse is false. Note that the -semimodule is easily seen to be a prime semimodule. Anyway, any direct summand of semimodule is subtractive, [11, page 184], hence is a subtractive subsemimodule of which is not dense in (see Example 2.6). Thus, is not a semimodule. One can ask when a prime semimodule can possibly be a semimodule. We will show later that, in the class of quasi-injective semimodule, the two concepts of semimodule and prime semimodule are equivalent.
It is well known that, for every -module , can be embedded in an injective -module. ̂ is called an injective hull of , if ̂ is an essential extension of , i.e for every non-zero submodule of ̂ [17].
It is well known, however, that injective hulls always exist if is a ring. But, Golan [10] proved that injective hulls of non-zero -semimodules need not exist for every semiring [10, prop.17.21, page 198]. If is a semiring then any cancellative -semimodule can be embedded in an injectivemodule ̂, [10, Ex.17.35, page 202]. Wang [19] proved that every semimodule over an additivelyidempotent semiring has an injective hull. For more details on an injective hull of semimodules over semiring, see for example information described previously [13]. Lemma 3.2. Let be a semisubtractive semiring, and let and be cancellative -semimodules. If and with ann ann , then defined by : is well-definedhomomorphism. Proof: Assume ́ , then either ́ , for some . Hence ́ ́ , ́ ́ , , ann( ), ann( ), , ́ ́ ́ . Or ́, by similar process ́ , ́ , and then is well-defined. On the other hand, it is clear that is -homomorphism.
Note that it is considered in this work that all semiring is a semisubtractive and all semimodules are cancellative. The following proposition gives another characterization of prime semimodules, which is analogous for modules [4]. Proposition 3.3. Let be a non-zero -semimodule having an injective hull ̂, then the following statements are equivalent: 1.
is a prime semimodule.

2.
is contained in every non-zero invariant subsemimodule of ̂. Proof: (1) (2) Let N be a non-zero invariant subsemimodule of ̂. We want to prove that . Since ̂ is an essential extension of , then . Thus . Since is prime, then , ann( ann( . We define as follows: , . By Lemma 3.2 we have that is a well-defined -hhomomorphism. Since ̂ is injective -semimodule then can be extended to ̂ ̂, as in the following diagram. where and are the inclusion -homomorphisms. Since is an invariant subsemimodule of ̂, then , but , then , hence . , which is a contradiction. Then ann( ) ann( ), and hence is a prime semimodule.
From Proposition 2.4, we have that every semimodule is a prime semimodule. Thus we have the following corollary. Corollary 3.4. Let be a semimodule having an injective hull ̂. If is a semimodule then is contained in every non-zero invariant submodule of ̂. Proposition 3.5. Let be a non-zero semimodule having an injective hull ̂. If is invariant subsemimodule of ̂ then the following statements are equivalent: 1.
has no non-trivial invariant subsemimodules. Proof: (1) (2). Let be a non-zero invariant subsemimodule of . Because is an invariant subsemimodule of ̂, so it can easily seen that is also invariant subsemimodule of ̂. Thus, by Proposition 3.3 we have , and hence . (2) (1). Let be a non-zero invariant subsemimodule of ̂. By Proposition 3.3, it is enough to show that . Since ̂ is an essential extension of , hence . Now we claim that is an invariant subsemimodule of . If this is proved, then by assumption has no non-trivial invariant subsemimodules and thus , which implies that . To prove the claim, consider any homomorphism in . Since , and since , so it is enough to show that . Because ̂ is an injective semimodule, then can be extended to ̂ ̂ , but is an invariant subsemimodule of ̂. Thus , hence is an invariant subsemimodule of . Now, as in the modules [7, page 22], we say that an -semimodule is said to be quasi-injective if each homomorphism from any subsemimodule into can be extended to a homomorphism of to . Note that any simple semimodule, and any injective semimodule, is quasi-injective. However, a quasi-injective semimodule needs not to be injective. For example, for each prime number , is considered as a -semimodule which is quasi-injective. In verity, the only non-zero subsemimodules of are , . Then, for each ,and all , the order of is less than or equal to , hence . It is clear that can be extended to a homomorphism in . Whereas, is not injective. The following theorem gives the relation between invariant and quasi-injective Semimodules.  where is the inclusion into the first factor and is the inclusion in the second factor. Suppose that can be extended to . Let be the natural projection, and let . It is easily seen that is a non-zero element in . But , which is a contradiction. This completes the proof. We conclude that is not an invariant subsemimodule of its injective hull ̂ . Thus we arrive at the following main theorem. Theorem 3.8. Let be any prime semimodule having an injective hull ̂. If is an invariant subsemimodule of ̂, then is a semimodule. Proof: We use the characterization of semimodules given in Proposition 2.11. So let be a nonzero stable subsemimodule of , then we have to show that is a contained in . From the definition of stability, it is easy to see that is invariant subsemimodule of . By assumption, is an invariant and prime semimodule and, using Proposition 3.5, has no non-trivial invariant subsemimodule. Therefore, . This completes the proof. The following corollary is immediate from Corollary 3.4 and Theorem 3.8. Corollary 3.9. Let be a semimodule having an injective hull ̂, if is an invariant subsemimodule of ̂. Then is a semimodule iff is a prime semimodule.
Next, similar to the case in the modules [20], we can say that an -semimodule is called compressible if every non-zero subsemimodule of contains an isomorphic copy of . As a trivial example:  Every simple -semimodule is compressible.  as a -semimodule is compressible.
̂ β  as a -semimodule is not compressible since . The following shows that the class of prime semimodules contains the class of compressible semimodules. Theorem 3.10. Every compressible -semimodule is a prime -semimodule. Proof: Let be a compressible -semimodule, and let . Now, we show that ann ann , since ann ann . So it is enough to prove that ann(N) ann . Since is compressible, then a monomorphism . Hence, ann , , thus which implies that , and , thus ann ann . This completes the proof.