Dedekind Multiplication Semimodules

The aim of this paper is to introduce the concept of Dedekind semimodules and study the related concepts, such as the class of semimodules, and Dedekind multiplication semimodules . And thus study the concept of the embedding of a semimodule in another semimodule.


Introduction
In ring theory, an ideal of a commutative ring with identity is said to be invertible if where and is the total quotient ring of . The concept of an invertible submodule was introduced by Naoum and Al-Alwan [1] as a generalization of the concept of an invertible ideal.
A semiring is a non-empty set together with two binary operations addition(+) and multiplication ( ) such that is a commutative monoid with identity element ; is a monoid with identity element ; for all ; and for every . We say that is a commutative semiring if the monoid is commutative. Let be an additive abelian monoid with additive identity . Then is called an -semimodule if there exists a scalar multiplication denoted by , such that ; ; ; and for all and all . Throughout this paper will denote a commutative semiring with identity, is unitary ( left) -semimodule. This paper consists four sections. Section 1 is devoted to introducing the concept of invertible subsemimodules of semimodule as a generalization of the concept of an invertible ideal in semiring. We will also find out some properties of this invertible subsemimodules. A non-zero ISSN: 0067-2904

Alwan and Alhossaini
Iraqi Journal of Science, 2020, Vol. 61, No. 6, pp: 1488-1497 1489 semimodule is a Dedekind semimodule if each non-zero subsemimodule of is invertible. Section 2 argues multiplication semimodules. We show that every multiplicatively cancellative multiplication semimodule is finitely generated. Section 3 discusses Dedekind multiplication semimodules. We show that if is a faithful multiplication -semimodule, then is a Dedekind semimodule iff is a Dedekind semiring. Let and be -semimodules, and . Here's a question that shows : when does contain a monomorphism?. If H contains a monomorphism we say that is embeds in .
It was proved by Low and Smith [2] that if is a torsionless multiplication -module then embeds in iff such that ann ann . Indeed if is not a multiplication semimodule then this condition is not sufficient see Remark 3.2. Here the importance of the invertible subsemimodules in obtaining the sufficient condition for the existence of a monomorphism.
In the last section we establish that if is any semimodule, with ⋂ and , and if there is a cyclic invertible subsemimodule in , then is a monomorphism.

Invertible Subsemimodules and Invertible Ideals
In this section we introduce the concept of invertible subsemimodule of a semimodule as a kind of generalization of the concept of invertible ideal in semiring. Remark (1.1): Let be a commutative semiring with identity 1. A set 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 equivalence relation on by , if for some . Then put R S the set of all equivalence classes of and define addition and multiplication on R S respectively by and , where also denoted by , we mean the equivalence class of . It is, then, easy to see that R S with the mentioned operations of addition and multiplication on R S in above is a semiring [3,4]. Definition (1.2): In Remark 1.1, 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 an -semimodule. If is a semidomain one can define the semifield of fractions ( ) of as the localization of at [5,6].

Definition (1.3):
Let be an -semimodule. In Remark 1.1, if is the set of all not zero-divisors of , and | for some implies . The total quotient semiring of the semiring is defined as the localization of at . Note that is also an -semimodule. Consider and ⁄ . Then {1} and so { }. Similar to that in modules see [1], we give the following remark. Remark (1.4): Let be an -semimodule and let be a non-zero subsemimodule of . Suppose that | Then is an -subsemimodule of , , and . Definition (1.5): Let be an -semimodule. A subtractive subsemimodule (or -subsemimodule) is a subsemimodule of such that if , then . A prime subsemimodule of is a proper subsemimodule of in which or whenever , [5]. We define -ideals and prime ideals of a semiring in a analogous manner [5]. Remark (1.6): Let be an -semimodule, we say that is a torsion-free semimodule whenever and with implies that either or . If is a subsemimodule of , then and ann are -ideals of , [5]. Proposition (1.7): Let be a non-zero -semimodule, and let be the set defined as in Definition 1.3, then has the following properties: 1) ⋂ann( ) is the empty set. 2) is a multiplicative subset of and .  [4] 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 not zero-divisor element such that . Let , be two fractional ideals of a semiring . Then . By Frac( ), we mean the set of all nonzero fractional ideals of a semiring . It is easy to check that Frac( ) equipped with the above multiplication of fractional ideals is an abelian monoid [4]. It is clear that each ideal of is fractional ideal of a semiring since (1) and (2) holds for , . Definition (1.9): [4] 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 will be denoted that by . The set of all invertible fractional ideals of is an abelian group. Example(1.10): Let be the set of all non-negative integers. Clearly its semifield of fractions. Let be a positive integer. The set is a fractional ideal of . It is clear as an -subsemimodule of is generated by and . While , where runs over all positive integers. Since there is no positive integer such that , is not a fractional ideal of . Let be a semidomain, its semifield of fractions, and -subsemimodules of . Then the residual quotient of by is defined as , see [6].

Proposition(1.11):
Let be a semidomain, and some fractional ideals of . Then the following statements hold: (1) .
is a fractional ideal of . , and so is an invertible ideal in if and only if is invertiblesubsemimodule in R R.
A semiring is semidomain if implies for all and all non-zero [6]. We say that a semidomain is said to be a Dedekind semidomain if every non-zero ideal of is invertible in [6]. According to the equivalent conditions explained on page 143 in Narkiewic book [7], a Dedekind domain is a domain in which non-zero fractional ideals form a group under multiplication. Inspired by this, we give the following definition: We define a semidomain to be a Dedekind semidomain if every non-zero fractional ideal of is invertible. Hence is a Dedekind semidomain if and only if Frac( ) is an abelian group. Corollary (1.23): Let be a semiring. Then 1) is Dedekind -semimodule if and only if is a Dedekind semidomain.

2)
is semimodule if and only if is a semidomain, i.e. each non-zero principal ideal of is invertible as a subsemimodule in if and only if it is generated by not a zero-divisor. The following remark shows that semimodule may not be D semimodule.

Remark (1.24):
Let be a semidomain, and the polynomial semiring in two independent variables and . Then is a semidomain. By Corollary 1.21, is a semimodule. But if we take the ideal generated by and , it is clear that is not invertible subsemimodule of . Thus is not a D semimodule. Next, we defined the notion of "essential" subsemimodule. In Golan book's [8], it was proposed the following definitions. An R-monomorphism of R-semimodules is essential if for any R-homomorphism , is a monomorphism implies that is a monomorphism.
A subsemimodule of an -semimodule is essential ( or large ) in if the inclusion mapping is an essential -monomorphism. Note that is an essential Rhomomorphism if and only if is a large subsemimodule of [8]. Another way for defining the notion of "essential" is proposed in [9] as follows. A subsemimodule of is said to be semi-essential in , written as , if for every subsemimodule of : . A monomorphism of -semimodules is said to be semi-essential if: . In [9], we have the following characterization of semi-essential subsemimodules. As a special case, we record the following. Corollary (1.30): If a semiring is a -semimodule. Then is a semidomain.

Multiplication Semimodules
In this section we study multiplication semimodules. We begin with following definition: Definition (2.
By Lemma 2.7, we have the following result. Corollary (2.11): Let be an entire yoked semiring and a cancellative faithful multiplication -semimodule. Then is finitely generated.
The next theorems give a characterization of multiplication semimodules, for the proof see [11]. Theorem(2.12): If is an multiplication -semimodule. Then is a projective -semimodule. Theorem (2.13): Let be a semidomain. If is an multiplication -semimodule, then is a torsion-free semimodule. Theorem (2.14): Let be a semidomain. If is an multiplication -semimodule, then is isomorphic to an invertible ideal in .

Dedekind Multiplication Semimodules
From Remark 2.3 we can say that a semiring is a Dedekind semidomain iff each non-zero ideal in is a multiplication ideal which contains a not zero-divisor. In this section we study Dedekind multiplication semimodules. We begin with the following.

Lemma (3.1): Let be a torsion-free -semimodule. If is an invertible subsemimodule of and
is an invertible ideal in , then is an invertible subsemimodule of .  [3,12].

Definition (3.3):
A semimodule is said to be duo if each subsemimodule of is invariant, [12].
In [12], we have the following characterization of duo subsemimodules. Theorem(3.4): Let be a yoked semidomain, and a torsion-free -semimodule. Then is duo if and only if for each -endomorphism of , there exists in such that for all . Remark(3.5): It is clear that any multiplication semimodule is duo. Hence by Theorem 3.4, if is a multiplication torsion-free semimodule over a yoked semidomain , then for each , , such that for all . Corollary (3.6): If is a torsion-free multiplication semimodule over a yoked semidomain , then there exists an epimorphism of semirings from onto . Proof: By Remark 3.5, , , such that and for all . Hence , defined by . It is easily check, that is an epimorphism of semirings. Theorem(3.7): If is a torsion-free multiplication semimodule over a yoked semidomain , then ann( ) Proof: By Corollary 3.6, | | | =ann( ). But , then ann( ). By Lemma 2.7, Theorem 2.13, and Theorem 3.7 we have.

Theorem(3.8):
If a cancellative faithful multiplication semimodule over a yoked semidomain . Then . The following lemma shows the importance of the faithful multiplication semimodules. Lemma(3.9): Let be a finitely generated cancellative faithful multiplication semimodule over a yoked semidomain . If is an invertible subsemimodule of for some ideal of , then is an invertible ideal in .. Proof: Since , then . By assumption , hence . It is clear that is an -subsemimodule of . Also, it is easy to see that every element of can be considered as an -endomorphism of . Now, since is a faithful multiplication semimodule, then by Theorem 3.8, . Therefore is an ideal in . As in modules see [13], it follows that . Hence , so which implies . Theorem (3.10): Let be a cancellative faithful multiplication -semimodule over a yoked Dedekind semidomain . Then is a finitely generated Dedekind -semimodule. Proof: Since is a faithful multiplication semimodule, and is a semidomain. By Corollary 2.11, we have is a finitely generated. Now, let be a non-zero subsemimodule of . Hence there exists a non-zero ideal in such that . Since is a Dedekind semidomain, thus is invertible in , and by Lemma 3.2, is invertible.
The following theorem is a converse of above theorem: Theorem (3.11): Let be a cancellative faithful multiplication semimodule over a yoked semidomain . If is a Dedekind semimodule, then is a Dedekind semidomain. Proof: By assumption, is a semidomain. By Corollary 2.11, we get is a finitely generated. Assume that is any non-zero ideal of . Then is a non-zero subsemimodule of , hence is invertible. From Lemma 3.9, is an invertible ideal.
A semidomain is said to be a Pr ̈fer semidomain if every non-zero finitely generated ideal of is invertible in [6]. Note that is a Dedekind semidomain if and only if is a Noetherian (each of its ideals is finitely generated) Pr ̈fer semidomain.
Let D be a Dedekind domain (D is a ring). By Theorem 3.7 in [4], the semiring of ideals Id(D) of D (the set of all ideals of D) is a Pr ̈fer semidomain. By Theorem 3.7 in [4], Id(D) is subtractive (each of its ideals is subtractive). If Id(D) is also Noetherian, then Id(D) is a Dedekind semidomain. Note that the semiring Id(D) is proper semiring, i.e., it is not a ring. If D is a Dedekind semidomain then the above argument remains true. Note that, each Noetherian Pr ̈fer semidomain is Dedekind.
For a more specific example, we assert that (Id( ),+, ) is a principal ideal semidomain (each of its ideals is principal) [6]. Hence, Id( ) is evidently a Dedekind semidomain. Note that the semiring (Id( ),+, ) is isomorphic to the semiring ( , gcd, ). Definition (3.12): A semimodule is said to be a Pr ̈fer semimodule if every non-zero finitely generated subsemimodule of is invertible in .
The proof of the following theorem is basically the same as the proof of the last results. Theorem (3.13): Let be a cancellative faithful multiplication semimodule over a yoked semiring . Then is a ̈ semimodule if and only if is a ̈ . If is a semimodule, we have the following remark which is special case of above theorem. Remark (3.14): Let be a cancellative faithful multiplication semimodule over a yoked semiring . Then is a semimodule if and only if is a . Proof:( ) By Corollary 1.29, we get is a semidomain, so each non-zero principal ideal is invertible.
( ) Assume that is a semidomain. Let now be a non-zero cyclic subsemimodule of , , for some ideal of . In this case we can take , and hence . By Corollary 2.11, we get is finitely generated, and thus is a multiplication ideal in [13]. But is a semidomain; thus by Theorem 2.3, is an invertible ideal in . Then by Lemma 3.2, is an invertible subsemimodule of .

Proposition (3.15):
If is a faithful multiplication Dedekind -semimodule. Then is also a faithful multiplication Dedekind -semimodule. Proof: Similarly in the proof of Theorem 3.10, is a f.g. faithful multiplication semimodule. So as in the modules see Corollary (2) of [2], we obtain that is a f.g. faithful multiplication -semimodule. By assumption and using Theorem 3.11, we get is a Dedekind semidomain. Now is a f.g. faithful multiplication -semimodule over the Dedekind semidomain , then by Theorem 3.10, is a Dedekind -semimodule.

Embedding of Semimodules
In this section we study "embeddability proplem" , thus we look for necessary and (or) sufficient conditions under which an -semimodule is isomorphic to a subsemimodule of the -semimodule . Now, put , is an -semimodule. We start by the following. Proposition (4.1): Let and be -semimodules. If there exists a monomorphism , then ann ann( ). Proof: It is clear that ann( ) ann( ), so it is enough to show that ann ann( ). Let ann , then . But is a monomorphism, therefore , and ann . But it is easily seen that ann ann , thus ann ann . Remark (4.2): The converse of Proposition 4.1 is not true in general. Proof: Let be a projective -semimodule with a non-commutative endomorphisms semiring, (for example can be any free semimodule of rank ˃1, such as as -semimodule). Put . Then , where and . If represents a generator of a semiring in the last direct sum, hence it is clear that ann ann . Whereas does not contain any monomorphism. To prove this, let such that . Thus is a projective ideal of (since is projective). And thus by [14], , so also is a multiplication ideal. By [15], is commutative. By [16, lemma 2.1], we have is commutative, which is a contradiction. Now, let us observe that if there exists a monomorphism , for any -semimodules, and , then it is clear that ⋂ . The following theorem gives a sufficient condition for the existence of a monomorphism in , in the case is a multiplication -semimodule.

Theorem(4.3):
Let be a multiplication -semimodule and any -semimodule such that ⋂ . Then for any , then is a monomorphism iff ann ann .

Proof:
If is a monomorphism then by Proposition 4.1, we have ann ann . Put . There is an ideal in such that . So , which implies ann . Then , hence , , and thus . Therefore and is a monomorphism. As a special case of Theorem 4.3, we give the following , comparison with [2, Lemma(1.1)]. We say that an -semimodule is called torsionless if ⋂ , . Corollary (4.4): Let be a torsionless multiplication -semimodule. Then is embeddable in iff such that ann ann . More generally, we have: Corollary (4.5): Let be a torsionless multiplication -semimodule. Then is embeddable in iff a f.g. subsemimodule of , which is generated by a set , where and ann ann .

Proof:
Assume that embeds in , i.e. which is a monomorphism. define as follows , where is the natural projection of onto the ith component. Note, since is isomorphic to the direct sum of copies of . Therefore ann ann and since is a monomorphism hence, by Proposition 4.1 ann ann . Now, ann ⋂ ann . Thus ann ann . Assume that a f.g. subsemimodule of , which is generated by a set , and ann ann . Now let us define an -homomorphism as follows ( ) . Now since ann ann , and by assumption ann ann ⋂ ann . Therefore by using Theorem 4.3, we obtain is a monomorphism in From our main results in this section, is that if such that ( ) is invertible in , and is torsionless, then is a monomorphism, and hence embeds in R, this means is isomorphic to an ideal of R. But now, let us recall that for any -semimodule , | if for some , then . Hence, for an -semimodule , | if for some , then . Theorem(4.6): Let and be any two -semimodules, with ⋂ , and . If there exists a cyclic invertible subsemimodule in , then is a monomorphism, and hence embeds in . Moreover, if ∑ , then is invertible subsemimodule in .