Isomorphism Theorems in the Primary Categories of Krasner Hypermodules

: Let R be a Krasner hyperring. In this paper, we prove a factorization theorem in the category of Krasner R -hypermodules with inclusion single-valued R -homomorphisms as its morphisms. Then, we prove various isomorphism theorems for a smaller category, i.e., the category of Krasner R -hypermodules with strong single-valued R -homomorphisms as its morphisms. In addition, we show that the latter category is balanced. Finally, we prove that for every strong single-valued R -homomorphism f : A → B and a ∈ A , we have Ker ( f ) + a = a + Ker ( f ) = { x ∈ A | f ( x ) = f ( a ) } .


Introduction
Algebraic hyperstructure theory addresses the study of algebraic objects endowed with multivalued operations, which are intended to generalize classical algebraic structures as groups, rings, or modules [1][2][3][4][5]. In the framework of that theory, hypergroups play a major role. A hypergroup is basically a set endowed by an associative multivalued binary operation, which fulfills an additional condition called reproducibility. Their inspection may reveal complex relationships among algebra, combinatorics, graphs, and numeric sequences [6,7]. Indeed, hyperstructures are inherently more complicated and bizarre than their classical counterparts. Hence, one of the main research directions in hyperstructure theory consists of identifying a subclass of a rather generic hyperstructure on the basis of a reasonable set of axioms, symmetries, or properties and proceeding with their analysis, in order to construct a theory that is at the same time general, profound, and beautiful.
Here, we consider one of the most important classes of hypergroups, that is the class of canonical hypergroups introduced by Mittas in [8]. This class constitutes the additive hyperstructure of many other hyperstructures, for example some types of hyperrings, hyperfields, and hypermodules. Notably, hyperrings and hyperfields, whose additive hyperstructure is a canonical hypergroup, were firstly introduced by Krasner [9]. Later, various authors defined and studied many other kinds of hyperrings and hypermodules; see, e.g., [5,[10][11][12]. In the context of canonical hypergroups, Madanshekaf [11] and Massouros [12] studied hypermodules whose additive structure is a canonical hypergroup equipped with a single-valued external multiplication. We call Krasner hypermodule a hypermodule equipped with a canonical hypergroup as its additive hyperstructure and an external single-valued multiplication, in order to distinguish it from other types of hypermodules. In fact, the name "Krasner" has been given to this kind of hypermodule in [13], inspired by the structure of the Krasner hyperring [9], even though such a hyperstructure has been previously considered by other authors in [11,12]. In fact, Krasner hypermodules are meant to generalize the concept of the Krasner hyperring.

Preliminaries
Throughout this paper, we use a few basic concepts and definitions that belong to standard terminology in hyperstructure theory. For more details about hyperstructures, the interested reader can refer to the classical references [1][2][3][4][5].
Let H be a non-empty set; let P(H) denote the set of all subsets of H; and let P * (H) = P(H) \ {∅}. Then, H together with a map: is called a hypergroupoid and is denoted by (H, +). The operation + is called the hyperoperation on H. Let A, B ⊆ H. The hyperoperation A + B is defined as: If there is no confusion, for simplicity {a}, A + {b}, and {a} + B are denoted by a, A + b, and a + B, respectively.

Definition 1.
A non-empty set S together with the hyperoperation +, denoted by (S, +), is called a semihypergroup if for all x, y, z ∈ S, (x + y) + z = x + (y + z), that is, the hyperoperation is associative.

Definition 2.
A semihypergroup (H, +) satisfying the reproducibility condition x + H = H + x = H for every x ∈ H is called a hypergroup.
Let e be an element of a semihypergroup (H, +) such that e + x = x for all x ∈ H. Then, e is called a left scalar identity. A right scalar identity is defined analogously. Moreover, an element x of a semihypergroup (H, +) is called a scalar identity if it is both a left and right scalar identity. A scalar identity in a semihypergroup H is unique, if it exists. In that case, we denote the scalar identity of H by 0 H . Let 0 H be the scalar identity of hypergroup (H, +) and

Definition 3.
A non-empty set M together with the hyperoperation + is called a canonical hypergroup if the following axioms hold: 1. (M, +) is a semihypergroup (associativity); 2. (M, +) is commutative (commutativity); 3. there is a scalar identity 0 M (existence of scalar identity); 4. for every x ∈ M, there is a unique element denoted by −x called the inverse of x such that 0 M ∈ x + (−x), which for simplicity, we write as 0 H ∈ x − x (existence of inverse); 5. for all x, y, z ∈ M, it holds that x ∈ y + z =⇒ y ∈ x − z (reversibility).
In the sequel, −x denotes the inverse of x in the hypergroup (M, +), and we write x − y instead of x + (−y). If there is no confusion, sometimes we omit the indication of the hyperoperation in a hypergroup, and for simplicity, we write M instead of (M, +). It is easy to verify that N ≤ M if and only if N = ∅ and x − y ⊆ N for all x, y ∈ N. Clearly, it follows that 0 M ∈ N. Hereafter, we recall some results discussed in [20] concerning the structure of the quotient of a canonical hypergroup with respect to a canonical subhypergroup.
Let (M, +) be a canonical hypergroup; let N be an arbitrary canonical subhypergroup of M; and set M N := {x + N | x ∈ M}. Consider the hyperoperation + on M N defined as: For notational convenience, we may writex instead of x + N.

Definition 5.
In every category C, 1. a morphism f : B → C is said to be a mono if for every g, h : A → B the following implication holds: 2. a morphism f : A → B is said to be an epi if for every g, h : B → C the following implication holds: 3. A morphism f : A → B of a category C is called an iso in C if there exists some g : In that case, g is denoted by f −1 .

Definition 6.
A non-empty set R together with the hyperoperation + and the operation · is called a Krasner hyperring if the following axioms hold: Definition 7. Let (R, +, ·) be a hyperring, and let S be a non-empty subset of R that is closed under the maps + and · in R. If S is itself a hyperring under these maps, then S is called a subhyperring of R. A subhyperring I of R is called a left (resp., right) hyperideal if r ∈ R and i ∈ I implies r · i ∈ I (resp., i · r ∈ I). A left and right hyperideal I is simply called a hyperideal of R.
We use the notation I R when the set I is a hyperideal of R. Moreover, we write I R if I R and I R, that is I is a proper hyperideal.
From now on, we overlook the name "Krasner" and simply use "hyperring". Furthermore, by R, we mean a (Krasner) hyperring. Usually, R is said to have a unit element if there exists y ∈ R such that x · y = y · x = x for every x ∈ R. If the element y exists, then it is unique, and we use the notation 1 R to denote it. In that case, we say that R is unitary. Definition 8. Let X and Y be two non-empty sets. A map * : X × Y → Y sending (x, y) to x * y ∈ Y is called a left external multiplication on Y. That operation is naturally extended to any U ⊆ X and V ⊆ Y by Analogously to the previous definition, a right external multiplication on Y is defined by * : Definition 9. Let (R, +, ·) be a hyperring. A canonical hypergroup (A, +) together with the left external multiplication * : R × A → A is called a left Krasner hypermodule over R if for all r 1 , r 2 ∈ R and for all a 1 , a 2 ∈ A, the following axioms hold:

Remark 1.
(i) If A is a left Krasner hypermodule over a Krasner hyperring R, then we say that A is a left Krasner R-hypermodule. Similarly, the right Krasner R-hypermodule is defined by the map * : A × R → A satisfying the similar properties mentioned in Definition 9 with affection on the right. (ii) If R is a hyperring with the unit element 1 R and A is a Krasner R-hypermodule satisfying 1 R * a = a (resp. a * 1 R = a) for all a ∈ A, then A is said to be a unitary left (resp. right) Krasner R-hypermodule.
(iii) From now on, for convenience, every hyperring R is assumed to have the unit element 1 R and by "an R-hypermodule A", we mean a unitary left Krasner R-hypermodule, unless otherwise stated.

Definition 10.
A non-empty subset B of an R-hypermodule A is said to be an R-subhypermodule of A, denoted by B ≤ A, if B is an R-hypermodule itself, that is for all x, y ∈ B and all r ∈ R, x − y ⊆ B and r * x ∈ B.

Remark 2.
We list here below some examples and properties related to R-subhypermodules, whose simple proofs are omitted for brevity: 1. Every hyperring R is an R-hypermodule, and every I R is an R-subhypermodule of R. 2. Let R be a hyperring. Every canonical hypergroup A can be considered as an R-hypermodule with the trivial external multiplication r * a = 0 A for every r ∈ R and a ∈ A. 3. Let A be an R-hypermodule and ∅ = B ⊆ A. For every I R, Definition 11. Let A and B be R-hypermodules. A function f : A → B that satisfies the conditions: for all r ∈ R and all x, y ∈ A, is said to be an (inclusion) R-homomorphism from A into B. Moreover, if the equality holds in Point 1, then f is called a strong (or good) R-homomorphism; and if f (x + y) ∩ f (x) + f (y) = ∅ holds instead of 1, then f is called a weak R-homomorphism.
The category whose objects are all R-hypermodules and whose morphisms are all R-homomorphisms is denoted by R hmod. The class of all R-homomorphisms from A into B is denoted by hom R (A, B). Moreover, we denote by R s hmod (resp., R w hmod) the category of all R-hypermodules whose morphisms are strong (resp., weak) R-homomorphisms. The class of all strong (resp., weak) R-homomorphisms from A into B is denoted by hom s R (A, B) (resp., hom w R (A, B)). It is easy to see that R s hmod is a subcategory of R hmod, and R hmod is a subcategory of R w hmod. Using standard notations, we express this as: The categories are usually called the primary categories of R-hypermodules; see, e.g., [19]. Now, let f ∈ hom R (A, B), and define: For further reference, we gather hereafter some information about Ker( f ) and Im( f ) from [13]: • Clearly, Im( f ) may not be an R-subhypermodule of B.

•
For every morphism f in R s hmod, Im( f ) is always an R-subhypermodule of the codomain of f .
Recall that if A is an R-hypermodule and B is a non-empty subset of A such that B is itself a hypermodule over R, then B is said to be an R-subhypermodule of B denoted by B ≤ A. Clearly, for every R-hypermodule A, {0 A } and A are two R-subhypermodules of A. In the following, we construct another R-hypermodule from A and B ≤ A called the quotient R-hypermodule.  (1) is an R-hypermodule with the external multiplication defined by: r (x + B) = r * x + B, for x, y ∈ A and r ∈ R.
Proof. As mentioned in Proposition 1, ( A B , + ) is a canonical hypergroup. First, we show that is well defined. Hence, let r 1 = r 2 ∈ R and x + B = y + B. We prove that r 1 * x + B = r 2 * y + B. In fact, since x ∈ y + b for b ∈ B, we have r 1 * x ∈ r 1 * y + r 1 * b ⊆ r 1 * y + B, so r 1 * x + B ⊆ r 1 * y + B. Consequently r 1 * x + B ⊆ r 2 * y + B. Analogously, r 1 * x + B ⊇ r 2 * y + B. Thus, r 1 (x + B) = r 2 (y + B) and is well defined.
Next, we check the axioms mentioned in Definition 9. Let r 1 , r 2 ∈ R and x + B, y + B ∈ A B . Clearly, imply that Axiom 4 of Definition 9 holds true. In order to prove the first axiom of Definition 9, note that: Furthermore, the second axiom of Definition 9 holds. Indeed,

Thus:
( Finally, as concerns the third axiom of Definition 9, we have: and the proof is complete.

Main Results
Factorization theorems are keystone results in abstract algebra. Indeed, these kinds of theorems relate the structure of two objects between which a homomorphism is given, in terms of the kernel and the image of the homomorphism. Moreover, they are preliminary steps toward more stringent results, where homomorphisms are replaced by isomorphisms. We start this section with a factorization theorem for an R-homomorphism between R-hypermodules. (A, B). If C is any R-subhypermodule included in Ker( f ), then there exists a unique R-homomorphism ψ ∈ hom R ( A C , B) such that f = ψ • π, where π : A → A C is the canonical quotient map. Hence, the diagram: . Clearly, ψ makes the diagram commute. Now, we need to check that it is a unique well-defined R-homomorphism.
A mono in the category R hmod (resp., R s hmod) is called an R-monomorphism (resp., strong R-monomorphism). An epi in the category R hmod (resp., R s hmod) is called an R-epimorphism (resp., strong R-epimorphism). An iso in the category R hmod (resp., R s hmod) is called an R-isomorphism (resp., strong R-isomorphism). When we say f : A → B is an R-isomorphism in R s hmod, automatically, f is a strong R-homomorphism.

Remark 3.
We point out some properties of R-isomorphisms in R hmod and R s hmod.
(i) An iso in the category R hmod (or an R-isomorphism) is surjective and injective, i.e., bijective. For this, let f : A → B be an iso in R hmod. Therefore, (ii) f : A → B is an R-isomorphism in R s hmod if and only if it is bijective. To show this fact, suppose f : A → B is bijective. Then, f −1 : B → A is also an R-homomorphism. Indeed, for every y 1 , y 2 ∈ B, there are (unique) x 1 , x 2 ∈ A such that f (x i ) = y i for i ∈ {1, 2}, and since f ( , and so, f −1 is an R-homomorphism. Therefore, f • f −1 = id B and f −1 • f = id A , and f is an R-isomorphism. The converse fact follows from (i) since every R-isomorphism in R s hmod is an R-isomorphism in R hmod.

Notation 1.
If there exists an iso between R-hypermodules A and B in R hmod (resp., R s hmod), we use the notation A ∼ = B (resp., A ∼ =s B). Moreover, if B ≤ A and x ∈ A, for convenience, we usex instead of x + B.
Proof. First note that Im(ϕ) is an R-subhypermodule of B. Let K = Ker(ϕ). Define a map f : Then, x ∈ȳ and x ∈ y + k for some k ∈ K. Hence, Therefore, ϕ(x) = ϕ(y). Thus, f (x) = f (ȳ), and the map f is well defined. If x, y ∈ A, then: On the other hand, Since x ∈ x + K and {a + K | a ∈ A} is a partition for A, we have x + K = y + K, i.e.,x =ȳ, and hence, f is injective. Clearly, f is surjective. Thus, according to the part (ii) of Remark 3, we have A K ∼ =s Im(ϕ), and the proof is complete.
Thus, g is a strong R-homomorphism. Now, x + A ∈ A+B A implies that x ∈ y + A for some y ∈ A + B. That is, y ∈ a + b for some a ∈ A, b ∈ B. Since y ∈ b + A, we get y + A = b + A by Lemma 1. Thus, g(b) = b + A = y + A = x + A, and g is surjective.
(⇒) Let f ∈ hom R (A, B) (resp., f ∈ hom s R (A, B)) be an R-monomorphism and f (x) = f (y). Then, f (r * x) = f (r * y) for an arbitrary r ∈ R. Now, define g, h ∈ hom s R (R, A) with g(r) = r * x and h(r) = r * y for each r ∈ R. Clearly,

Proof.
(i) Let f ∈ hom R (A, B) be surjective, and let g, h ∈ hom R (B, C). If g • f = h • f , then for all a ∈ A, we have g( f ((a)) = h( f (a)). Now, let b ∈ B. Clearly, there is x ∈ A such that f (x) = b. Thus, . Hence, f is an R-epimorphism. (ii) (⇐) It is clear from (i).
(⇒) Let f ∈ hom s R (A, B) be an R-epimorphism and b ∈ B. Suppose f is not surjective. Then, This contradiction shows that f is surjective.
A morphism is said to be a bimorphism if it is a mono, as well as an epi. A category is said to be balanced when a morphism is a bimorphism if and only if it is an iso. Then, f −1 ( f (a)) = Ker( f ) + a = a + Ker( f ).

Conclusions
Krasner hypermodules have been already considered from the standpoint of category theory by various authors in, e.g., [11][12][13]15], focusing on the properties of different types of homomorphisms, notably the so-called inclusion homomorphisms, strong homomorphisms, and weak homomorphisms, according to their behavior with respect to the multivalued addition. Since morphisms play an important role in every category, one needs a clear understanding of fundamental theorems concerning homomorphisms, such as factorization and isomorphism theorems, in order to pursue fundamental studies in category theory. In this paper, we first studied the primary categories of Krasner hypermodules over a Krasner hyperring, introduced in [19]. In particular, we proved factorization and isomorphism theorems regarding both inclusion and strong single-valued homomorphisms as their morphisms. Moreover, we focused on strong isomorphism theorems between quotient hypermodules, and we showed that the category of Krasner R-hypermodules with strong single-valued homomorphisms is balanced. Arguably, analogous results may have a different and more complex form in other categories of (general) Krasner hypermodules, depending on multivalued homomorphisms. On the basis of preliminary results presented in [13,15,19], we believe that the exploration of factorization and isomorphism theorems for multivalued homomorphisms between hypermodules is a possible direction for further research.