International Journal of Analysis and Applications Volume 16, Number 6 (2018), 793-808 URL: https://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-16-2018-793 VARIOUS KINDS OF FREENESS IN THE CATEGORIES OF KRASNER HYPERMODULES HOSSEIN SHOJAEI∗ AND REZA AMERI School of Mathematics, Statistics and Computer Science, College of Science, University of Tehran, P.O. Box 14155-6455, Tehran, Iran ∗Corresponding author: h shojaei@ut.ac.ir Abstract. The purpose of this paper is to study the concept of freeness in the categories of Krasner hypermodules over a Krasner hyperring. In this regards first we construct various kinds of categories of hypermodules based on various kinds of homomorphisms of hypermodules, such as homomorphisms, good homomorphisms, multivalued homomorphisms and etc. Then we investigate the notion of free hypermodule in these categories. This leads us to introduce different types of free, week free, �∗-free and fundamental free hypermodules and obtain the relationship among them. 1. Introduction The concept of hyperstructure is the generalization of the concept of algebraic structure. As a matter of fact, the hyperstructures are more natural and general than the algebraic structures. For the first time, hypergroups, as a suitable generalization of groups, were defined by Marty in 1934 [11]. Recently, many hyperstructures, for example, hypergroups, hyperrings, hyperfield, hypermodules and hypervector spaces, have been introduced and studied by many authors, e.g., [1], [3], [4], [5], [6], [7], [9], and [13]. For the first time, the concepts of hyperring and hyperfield were introduced by Krasner in connection with his work on valued fields. One of the most important hyperstructures satisfying the module-like axioms as a generalization of module is a type of hypermodule over a Krasner hyperring that we call it Krasner hypermodule (see [14]). Received 2017-11-07; accepted 2018-01-17; published 2018-11-02. 2010 Mathematics Subject Classification. 18D35. Key words and phrases. Krasner hyperring; Krasner hypermodule; free hypermodule. c©2018 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 793 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-16-2018-793 Int. J. Anal. Appl. 16 (6) (2018) 794 Interested readers can find a wide generalization of Krasner hypermodules in [15]. In this paper, we study these hyperstructures and focus on various freeness for a Krasner hypermodule. Next Section is a summary and reminder of [14]. 2. Preliminaries We start this section with some basic and fundamental concepts of category theory, and then we proceed to recall some requirements from hyperstructures theory. Definition 2.1. A category denoted by C consists of (1) A class of objects: A,B,C,. . . (2) A class of morphisms or arrows: f,g,h, . . . with the following data: • Given morphisms f : A −→ B and g : B −→ C, that is, with: cod(f) = dom(g) there is given a morphism: g ◦f : A −→ C called the composition of morphisms f and g. • Associativity: h◦ (g ◦f) = (h◦g) ◦f for all f : A −→ B, g : B −→ C and h : A −→ C. • Identity: f ◦ idA = f = idB ◦f for all f : A −→ B. Notation 2.1. The classes of objects and morphisms of a category are denoted by Ob(C) and Mor(C), respectively. The class of all morphisms from A to B of category C is denoted by MorC(A,B). Note that a morphism f : A −→ B in 2.1 is not necessarily a function from A to B. Recall that a zero object in an arbitrary category C is an object denoted by 0 such that |MorC(M, 0)| = |MorC(0,M)| = 1 for every M ∈ Ob(C). The morphism A −→ 0 −→ B of MorC(A,B) in a category C is called the zero morphism (see [2] or [8]). Throughout this paper, P(X) denotes the set of all subsets of X and P∗(X) = P(X) \ {∅}. Here, Sets denotes the category of sets as objects with functions between sets as morphisms. Now we state some basic definitions related to hyperstructures theory. Let H be a non-empty set. Then H together with the map · : H ×H −→ P∗(H) (a,b) 7→ a · b denoted by (H, ·) is called a hypergroupoid and · is called a hyperproduct or hyperoperation on H. Let A,B ⊆ H. The hyperproduct A ·B is defined as A ·B = ⋃ (a,b)∈A×B a · b. Int. J. Anal. Appl. 16 (6) (2018) 795 If there is no confusion, for simplicity {a}, A ·{b} and {a}·B are denoted by a, A ·b and a ·B, respectively. Also we use ab instead of a · b for a,b ∈ H. 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). A semihypergroup (H, ·) satisfying x ·H = H ·x = H for every x ∈ H, is called a hypergroup. Let x be an element of semihypergroup (H, +) (resp., (H, ·)) such that e+y = y+e = y (resp., e·y = y·e = y). Then x is called a scalar identity (resp., unit). Every scalar identity or scalar unit in a semihypergroup H is unique. We denote the scalar identity (resp., unit) of H by 0H (resp., 1H). Let 0H (resp., 1H) be the scalar identity (resp., unit) of hypergroup (H, +) (resp., (H, ·)) and x ∈ H. An element x′ ∈ H is called an inverse of x in (H, +) (resp., (H, ·)) if 0H ∈ x+x′∩x′+x (resp., 1H ∈ x·x′∩x′·x). A semihypergroup with a scalar identity is called a hypermonoid. A non-empty set H together with the hyperoperation + is called a canonical hypergroup if the following axioms hold: (1) (H, +) is a semihypergroup (associativity); (2) (H, +) is commutative (commutativity); (3) there is a scalar identity 0H (existence of scalar identity); (4) for every x ∈ H, there is a unique inverse denoted by −x such that 0H ∈ x + (−x), which for simplicity we write 0H ∈ x−x (existence of inverse); (5) ∀x,y,z ∈ H : x ∈ y + z =⇒ y ∈ x−z (reversibility). Definition 2.2. A non-empty set R together with the hyperoperation + and the operation · is called a Krasner hyperring if the following axioms hold: (1) (R, +) is a canonical hypergroup; (2) (R, ·) is a semigroup including 0R as a bilaterally absorbing element, that is 0R ·x = x · 0R = 0R for all x ∈ R; (3) (y + z) ·x = (y ·x) + (z ·x) and x · (y + z) = x ·y + x ·z for all x,y,z ∈ R. We say R has a unit (element) 1R when 1R ·r = r · 1R = r for all r ∈ R. Example 2.1. Let (R, +, ·) be a ring and N a normal subgroup of semigroup (R\{0}, ·). Let R′ = R N be the set of classes of the form x̄ = x ·N. If for all x̄, ȳ ∈ R̄, we define x̄ +′ ȳ = {z̄| z ∈ x̄ + ȳ}, and x̄ ·′ ȳ = x ·y, then (R, +′, ·′) is a Krasner hyperring. Int. J. Anal. Appl. 16 (6) (2018) 796 Now we state the concept of hypermodule over a hyperring. One of the most important and well-behaved classes of hypermodules is a class induced by the structure of a Krasner hyperring. To study such hyper- structures, we start with the following concept. Definition 2.3. 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 . If U ⊆ X and V ⊆ Y , then we define u∗V := ∪v∈V u∗v and U ∗v := ∪u∈Uu∗v. Analogously, a right external multiplication on Y is defined by ∗ : Y ×X −→ Y sending (y,x) to y∗x ∈ Y . Definition 2.4. Let (R, +, ·) be a Krasner hyperring. A canonical hypergroup (A, +) together with the left external multiplication ∗ : R×A −→ A on A is called a left Krasner hypermodule over R if for all r1,r2 ∈ R and for all a1,a2 ∈ A the following axioms hold: (1) r1 ∗ (a1 + a2) = r1 ∗a1 + r1 ∗a2; (2) (r1 + r2) ∗a1 = r1 ∗a1 + r2 ∗a1; (3) (r1 ·r2) ∗a1 = r1 ∗ (r2 ∗a1); (4) 0R ∗a1 = 0A. Remark 2.1. (i) If A is a left Krasner hypermodule over a Krasner hyperring R, then we say that A is a left Krasner R-hypermodule. Clearly, a right Krasner R-hypermodule is defined with the map ∗ : A×R −→ A possessing the smilar properties. (ii) If R is a Krasner hyperring with 1R and A is a Krasner R-hypermodule satisfying 1R ∗a = a (resp. a∗ 1R = a) for all a ∈ A, then A is said a unitary left (resp. right) Krasner R-hypermodule. (iii) Throughout the paper, for convenience, by hyperring R we mean a Krasner hyperring with 1R and by R-hypermodule A we mean a unitary left Krasner R-hypermodule unless otherwise stated. Definition 2.5. A non-empty subset B of an R-hypermodule A is said to be an R-subhypermodule of 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. Proposition 2.1. [14, Remark 3.2] (i) In every (unitary) R-hypermodule A, (−1R) ∗a = −a for every a ∈ A. (ii) In every hyperring R, (−1R) ·r = −r for every r ∈ R. Remark 2.2. Let A be an R-hypermodule, r,s ∈ R and a ∈ A. In the sequel, when there is no confusion, we use rs and ra instead of r ·s and r ∗a, respectively. Unlike the category of modules, there are various types of homomorphisms in the categories of hypermod- ules. Int. J. Anal. Appl. 16 (6) (2018) 797 Definition 2.6. Let A and B be two R-hypermodules. A function f : A −→ B that satisfies the conditions: (i) f(x + y) ⊆ f(x) + f(y); (ii) f(rx) = rf(x); for all r ∈ R and all x,y ∈ A, is said to be an (inclusion) R-homomorphism from A into B. Remark 2.3. If in (i) of Definition 2.6 the equality holds, then f is called a strong (or good) R-homomorphism. The category whose objects are all R-hypermodules and whose morphisms are all R-homomorphisms is denoted by Rhmod. The class of all R-homomorphisms from A into B is denoted by homR(A,B). Also, Rshmod is the category of all R-hypermodules whose morphisms are strong R-homomorphisms. The class of all strong R-homomorphisms from A into B is denoted by homsR(A,B). It is easy to see that Rshmod is a subcategory of Rhmod, and we write Rshmod �Rhmod and read Rshmod is a subcategory of Rhmod. So far we have considered the morphisms or arrows, as usual, the functions between objects. But one can consider a morphism from A to B as a function from A into P∗(B) called a multivalued function from A to B. Considering multivalued functions between sets, we have the following definition: Definition 2.7. Category of hypersets denoted by HSets is a category with the following data: (1) Ob(HSets) = Ob(Sets), (2) Mor(HSets) = the class of all multivalued functions between objects, that the composition g ◦f is defined as the following: (g ◦f)(a) = ⋃ b∈f(a) g(b), ∀a ∈ A, (2.1) and an identity morphism for an object A is idA(x) = {x} for all x ∈ A. Now we are ready to define a generalization of usual morphisms of Rhmod. Definition 2.8. If A and B are two R-hypermodules, then multivalued function f from A into B is a mapping f : A −→ P∗(B) satisfying the following conditions: (i) f(x + y) ⊆ f(x) + f(y); (ii) f(rx) = rf(x); for all r ∈ R and all x,y ∈ A, is said to be a multivalued R-homomorphism, for short Rmv-homomorphism. Remark 2.4. In (i) of Definition 2.8, if the equality holds, then f is called a strong (or good) multivalued R-homomorphism, for short an Rsmv-homomorphism. Notation 2.2. The class of all Rmv-homomorphisms (resp., Rsmv-homomorphisms) from A into B is denoted by HomR(A,B) (resp., Hom s R(A,B)). Int. J. Anal. Appl. 16 (6) (2018) 798 Proposition 2.2. [14, Remark 3.9] (i) For every f ∈ homR(A,B), f(0A) = 0B. (ii) For every f ∈ homR(A,B), f(−x) = −f(x) and f(x−y) = f(x) −f(y). Let f ∈ HomR(A,B) and h ∈ HomR(B,C). The composition h◦f is defined as Equation 2.1. Also, for every R-hypermodule A, the R-homomorphism idA with definition idA(x) = {x} for all x ∈ A is the identity morphism as before. Hereafter, RHmod (resp., RsHmod) denotes the category whose objects are all R-hypermodules and whose morphisms from A to B are all Rmv-homomorphisms (resp., Rsmv- homomorphisms) from A into B. Clearly, RsHmod is a subcategory of RHmod, i.e., RsHmod �RHmod. Remark 2.5. (i) Hereafter, we identify a singleton X = {a} by its element a. Also, we sometimes write f(a) = b instead of f(a) = {b}. So every single-valued morphism f ∈ HomR(A,B) (resp., f ∈ HomsR(A,B)) is an element of homR(A,B) (resp., hom s R(A,B)), and conversely, every element of homR(A,B) (resp., hom s R(A,B)) can be considered as an element of HomR(A,B) (resp., Hom s R(A,B)), So Rhmod �RHmod (resp., Rshmod �RsHmod). (ii) Let f,g ∈ HomR(A,B). Define the relation ≤ on HomR(A,B) in which f ≤ g means f(x) ⊆ g(x) for all x ∈ A. Clearly (HomR(A,B),≤) is a poset. For convnience and distinguishing, we call Rhmod and Rshmod primary categories of Krasner R- hypermodules. Also, RHmod and RsHmod are called secondary categories of Krasner R-hypermodules. 3. Freeness of hypermodules As it is well-known free objects play an important role in the study of modules theory. In [12] it was shown that free object does not exist in the category of hypergroups. Also, in [10] the notion of free hypermodules in the category of Krasner hypermodules was introduced. However, it is not clear that whether this definition is suitable in view point of category theory. Here we give various types of freeness in the categories of R-hypermodules and investigate the relationship between them. Fix a hyperring (R, +, ·). Let U(R) denote the set of all expressions of the form ∑ i∈I ( ∏ j∈Ji rj) in which rj ∈ R where I and all Ji’s are finite. The relation γ is defined on R is defined as follows: for all x,y ∈ R, xγy ⇐⇒∃u ∈U(R) : x,y ∈ u. The transitive closure of the relation γ is called the fundamental relation of R denoted by γ∗. Let γ∗(r) denote the equivalence class containing r ∈ R. Then it is shown that R γ∗ with the sum ⊕ and product ⊗ is a ring as follows: Int. J. Anal. Appl. 16 (6) (2018) 799 for all x,y ∈ R, γ∗(x) ⊕γ∗(y) = γ∗(z) ∀z ∈ γ∗(x) + γ∗(y); γ∗(x) ⊗γ∗(y) = γ∗(x ·y). The fundamental relation γ∗ is the smallest equivalence relation such that R γ∗ is a ring. The ring R γ∗ is called the fundamental ring of R. Also, the fundamental relation of an R-hypermodule A can be defined similar to above denoted by �∗A that A �∗ A is a fundamental module over the ring R γ∗ with operations: �∗A(x) ⊕ � ∗ A(y) = � ∗ A(z) ∀z ∈ � ∗ A(x) + � ∗ A(y); γ∗(r) � �∗A(x) = � ∗ A(r ∗x), for all x,y ∈ A and r ∈ R. The fundamental relation �∗A is the smallest equivalence relation such that A �∗ A is a module over the ring R γ∗ . (For more details, see [16] and [17]). Now we introduce the following concept: Definition 3.1. An R-hypermodule F is said to be free on X ⊆ F if for every R-hypermodule A and for any morphism f : X −→ A in HSets, there exists a unique f̄ ∈ HomR(F,A) such that f̄ ◦ i = f in which i = idF |X, i.e., the following diagram commutes: X i // f �� F ∃!f̄~~ A Remark 3.1. In [10] Ch. G. Massouros defined a free R-hypermodule differently. In the sense of Massouros, an R-hypermodule F is said to be free on X ⊆ F if X generates F (see Definition 3.5) and for every R- hypermodule A and for an arbitrary morphism f : X −→ A in Sets, there exists f̄ ∈ HomR(F,A) such that f̄(x) = {f(x)} for every x ∈ X, i.e., the following diagram commutes: X i // f �� F ∃f̄~~ A By this definition, the morphism f̄ is not necessarily unique Rmv-homomorphism, but in [10], it was shown that f̄ is a maximum Rmv-homomorphism, that is f̄ is a maximum element in the poset (HomR(F,A),≤)) such that f̄ ◦ i = f. Clearly, a free R-hypermodule on X ⊆ F based on this definition, is not really free on X in RHmod or Rhmod. In the following, we introduce the concept of weak freeness over HSets or Sets motivated by Massouros definition. Int. J. Anal. Appl. 16 (6) (2018) 800 Definition 3.2. An R-hypermodule F is said to be weak free on X ⊆ F over HSets (resp., Sets) if for every R-hypermodule A and for every morphism f : X −→ A in HSets (resp., Sets), there exists a maximum f̄ ∈ HomR(F,A) such that f̄ ◦ i = f in which i = idF |X, i.e., the diagram X i // f �� F ∃f̄~~ A commutes. Remark 3.2. (i) Every free R-hypermodule on X is weak free on X over HSets. (ii) Every free R-hypermodule on X in the sense of Massouros is really weak free on X over Sets. Notation 3.1. Let A be an R-hypermodule and X ⊆ A and set �∗A(X) := ∪x∈X� ∗ A(x). For X = {x}, we write �∗A(x) instead of � ∗ A({x}). Definition 3.3. Two morphisms f and g in HomR(A,B) is said to be � ∗-equivalent, and write f ∼�∗ g if and only if �∗B(f(x)) = � ∗ B(g(x)) for every x ∈ A. Clearly ∼�∗ is an equivalence relation on HomR(A,B). We denote the equivalence class of f with respect to ∼�∗ by [f]. Definition 3.4. An R-hypermodule F is said to be �∗-free on X ⊆ F over HSets (resp., Sets) if for every R-hypermodule B and for every morphism f : X −→ B in HSets (resp., Sets), there exists an �∗-unique f̄ ∈ HomR(F,B) such that f̄ ◦ i = f in which i = idF |X, i.e., if there exists f̄′ ∈ HomR(F,B) such that f̄′ ◦ i = f, then [f] = [g]. Now we recall the notion of a generating set. Definition 3.5. Let R be a hyperring not necessarily with 1R and A be an R-hypermodule and X ⊆ A. 〈X〉 denotes the smallest R-subhypermodule of A containing X or the intersection of all R-subhypermodules of A containing X. Notation 3.2. Let A be an R-hypermodule. Then for x ∈ A and m ∈ Z, mx =   x + x + · · · + x︸ ︷︷ ︸ m times x if n > 0 0A if n = 0 −x−x−···−x︸ ︷︷ ︸ −m times x if n < 0. Int. J. Anal. Appl. 16 (6) (2018) 801 Proposition 3.1. For a subset X of a not necessarily unitary R-hypermodule A, 〈X〉 is the set {a ∈ m∑ i=1 rixi + n∑ i=1 nixi + k∑ i=1 ki(xi − xi) | ri ∈ R, xi ∈ X, m,n,k,ki ∈ N, ni ∈ Z}. Proof. It is clear to straightforward. � Definition 3.6. The set X is said to be a generating set for an R-hypermodule A, or X generates A, if A = 〈X〉. Here, A is called finitely generated if it has a finite generating set. Let X = {x}. For simplicity, we use 〈x〉 instead of 〈X〉. It is easy to see that 〈x〉 = {a ∈ rx + mx + n∑ i=1 ni(x−x) | r ∈ R, m ∈ Z, n,ni ∈ N}. Let Rx = {rx | r ∈ R, x ∈ A}. Remark 3.3. Let R be with identity 1R and A be a unitary R-hypermodule. Then (i) 〈x〉 = Rx. Indeed, since mx = x + x + · · · + x︸ ︷︷ ︸ m times x = 1R ∗x + 1R ∗x + · · · + 1R ∗x︸ ︷︷ ︸ m times x = (1R + 1R + · · · + 1R︸ ︷︷ ︸ m times 1R ) ∗x ⊆ Rx and (by Proposition 2.1 (i)) x−x = x + (−x) = 1R ∗x + ((−1R) ∗x) = (1R + (−1R)) ∗x = (1R − 1R) ∗x ⊆ Rx, we have 〈x〉 = Rx. (ii) Letting X = {xi}i∈I ⊆ A, A = 〈X〉 if and only if for every a ∈ A, there exists a finite J ⊆ I such that a ∈ ∑ j∈J rjxj which rj ∈ R and xj ∈ X. Definition 3.7. Let A be an R-hypermodule and X ⊆ A. X is said linearly independent if for all n ∈ N and all x1,x2, . . . ,xn ∈ X, 0A ∈ n∑ i=1 rixi implies r1 = r2 = · · · = rn = 0R. Definition 3.8. Let F be an R-hypermodule and X be a generating set which is linearly independent. Then X is called a basis for F . Remark 3.4. The empty set is linearly independent and is a basis for the trivial R-hypermodule 0 = {0} (based on Definition 3.5). If X is a basis for F, then for every a ∈ F there are x1,x2, . . . ,xn ∈ X and unique r1,r2, . . . ,rn ∈ R such that a ∈ n∑ i=1 rixi. In order to show the uniqueness of ri’s, let a ∈ m∑ i=1 r′ixi for some m ∈ N. Without loss of generality, assume m = n. Then 0F ∈ a−a ∈ n∑ i=1 rixi − n∑ i=1 r′ixi Int. J. Anal. Appl. 16 (6) (2018) 802 =⇒ 0F ∈ n∑ i=1 (ri −r′)xi =⇒∃ci ∈ ri −r′i : 0F ∈ n∑ i=1 cixi Since X is a basis, ci = 0R. So 0R ∈ ri −r′i implies ri = r ′ i. Every ri (i = 1, 2, . . . ,n) is called the ith coordinate of a in R. In fact, every coordinate of a can be considered as a function from F into R mapping a to an appropriate ri denoted by fi. Indeed, fi(xj) = 1R if i = j, fi(xj) = 0R if i 6= j and fi(a) = ri for a ∈ n∑ i=1 rixi. Every fi is called ith coordinating function of a. Clearly for all a,b ∈ F and all r ∈ R, we have fi(a + b) ⊆ fi(a) + fi(b) and fi(ra) = rfi(a). For an arbitrary morphism f : X −→ B in HSets, define the morphism f̄ : F −→ B f̄(a) = ∑ i fi(a) ∗f(xi) (3.1) that the summation is indeed taken finite for an appropriate n, and fi(a) = ri is the ith coordinate of a. Since X is a basis and the ith coordinate of a is uniquely determined, so f̄ is well-defined. It is easy to see that f̄(xi) = f(xi) for all xi ∈ X. f̄(a + b) = ⋃ c∈a+b f̄(c) = ⋃ c∈a+b (∑ i (fi(c) ∗f(xi)) ) (by Equation 3.1) ⊆ ∑ i ( ⋃ c∈a+b {fi(c)} ) ∗f(xi) ⊆ ∑ i fi(a + b) ∗f(xi) ⊆ ∑ i fi(a) ∗f(xi) + ∑ i fi(b) ∗f(xi) = f̄(a) + f̄(b) (by Equation 3.1). Note that the first inclusion is obtained from fi(a + b) ⊆ fi(a) + fi(b). Clearly f̄(ra) = rf̄(a) for all r ∈ R. So f is an Rmv-homomorphism and thus f̄ ∈ HomR(F,B). Now let f̄′ ∈ HomR(F,B) be another Rmv- homomorphism with f̄′(xi) = f̄(xi) for every xi ∈ X. Then for every a ∈ F, we have a ∈ ∑ i fi(a) ∗xi and thus f̄′(a) ⊆ f̄′ (∑ i (fi(a) ∗xi) ) = ∑ i fi(a) ∗ f̄′(xi) = ∑ i fi(a) ∗f(xi) = f̄(a). (3.2) Hence f̄′ ≤ f̄. Thus we have the following statement: Int. J. Anal. Appl. 16 (6) (2018) 803 Theorem 3.1. Let F be an R-hypermodule with basis X. Then F is weak free on X over HSets. Theorem 3.2. Let F be an R-hypermodule with basis X. If F is weak free on X over Sets, then it is �∗-free on X over Sets. Proof. Following the proof of Theorem 3.1, consider Equation 3.2. If f is a morphism of Set, then f̄′(a) and f̄(a) are contained in the finite linear combination ∑ i fi(a) ∗f(xi) in which fi(a) ∈ R and f(xi) ∈ B. Thus �∗B(f̄ ′(a)) = �∗B(f̄(a)). Hence F is � ∗-free on X. � Definition 3.9. If A = ∑ i∈I Ai in which every Ai is an R-subhypermodule of A and Aj ∩ ∑ i 6=j Ai = {0A} for every j (j ∈ J), then we write A = ⊕i∈IAi, and A is said to be the direct sum of {Ai}i∈I. Proposition 3.2. Let X = {xi}i∈I be a subset of an R-hypermodule F which I is an index set. If for every a ∈ F , there exist a finite set I0, unique rj ∈ R and xj ∈ X such that a ∈ ∑ j∈I0 rjxj, then F = ⊕i∈IRxi. Proof. Suppose every element of F is contained in a uniquely expressed linear combination of the form n∑ i=1 rixi in which ri ∈ R and xi ∈ X for an appropriate n ∈ N. Consequently, rxi = 0F for r ∈ R implies r = 0R. Also, for every element a ∈ F , a ∈ n∑ i=1 Rxi for some n ∈ N. So F = ∑ i∈I Rxi. Suppose a ∈ Rxj and a ∈ ∑ i 6=j Rxi that the summation is taken finite. So assume a = rjxj and a ∈ n∑ i=1,i6=j rixi in which r1,r2, . . . ,rn ∈ R for an appropriate n ∈ N by a new indexing. Thus 0F ∈ a−a ∈   n∑ i=1,i6=j rixi  − (rjxj). But −(rj ∗xj) = (−1R) ∗ (rj ∗xj) = (−1R ·rj) ∗xj = (−rj) ∗xj, from Proposition 2.1. So 0F ∈ ( n∑ i=1,i6=j rixi ) + (−rj)xj. Since X is a basis for F , we obtain −rj = 0R = ri for all 1 ≤ i ≤ n and i 6= j. Consequently a = 0F . � Corollary 3.1. Let F be an R-hypermodule with basis X. Then F = ⊕i∈IRxi. Proof. It is clear from Proposition 3.2. � The following result shows that the converse of Proposition 3.2 holds: Indeed, Theorem 3.3. Let X = {xi}i∈I be a subset of an R-hypermodule F which I is an index set. For every a ∈ F , there exist a finite set I0, unique rj ∈ R and xj ∈ X such that a ∈ ∑ j∈I0 rjxj if and only if F = ⊕i∈IRxi. Int. J. Anal. Appl. 16 (6) (2018) 804 Proof. According to Proposition 3.2, we must suppose F = ⊕i∈IRxi and prove for every a ∈ F, there exist a finite set I0, unique rj ∈ R and xj ∈ X such that a ∈ ∑ j∈I0 rjxj. Clearly a ∈ F = ⊕i∈IRxi implies that there exist n ∈ N, rj ∈ R and xj ∈ X such that a ∈ n∑ j=1 rjxj. To show the uniqueness of rj ∈ R, let a ∈ m∑ j=1 r′jxj for some m ∈ N. Without loss of generality, n = m. Then 0F ∈ a−a ∈ n∑ j=1 rjxj − n∑ j=1 r′jxj. Thus 0F ∈ n∑ j=1 djxj where dj ∈ rj − r′j. If dj = 0R for every 1 ≤ j ≤ n, then by the reversibility of R, rj = r ′ j for every 1 ≤ j ≤ n. Without loss of generality, suppose d1 6= 0R and d1x1 6= 0F . Then 0F ∈ n∑ j=1 djxj, and thus d1x1 ∈ n∑ j=2 djxj by the reversibility of F. So d1x1 ∈ Rxi∩ n∑ j=2 Rxj that is a contradiction. Hence the proof is complete. � Theorem 3.4. Given a set X, there exists an R-hypermodule F and some Y ⊆ F with |X| = |Y | such that F is weak free on Y over HSets. Proof. Clearly, R can be regarded as an R-hypermodule, and so, one can form the direct sum F = ⊕i∈XRi, where for all i ∈ X, Ri = R. Define l: X −→ F as follows: l(x) = (ri,x)i∈X where ri,x = δi,x. It can be easily shown that {(ri,x)i∈X | x ∈ X} is a basis for F. Denote (ri,x)i∈X as ex. Then we can write F = ⊕x∈XRex and every element of Fis contained in a unique finite linear combination ∑ x∈X rxex where rx ∈ R. Indeed, every element of F has the form (rx)x∈X in which all but only a finitely many rx’s are zero. So the subset {ex}x∈X is a basis for F . Thus by Theorem 3.1, F is weak free on {ex}x∈X. Consequently, considering the injective map l: X −→ F with x 7→ ex and letting Y = l(X), F is weak free on Y . � Theorem 3.5. For every R-hypermodule A, there is some surjective f̄ ∈ HomR(F,A) in which F is weak free R-hypermodule on some Y ⊆ F over HSets. Proof. Let A = 〈X〉. Also, let F, Y ⊆ F and l: X −→ Y as in the proof of Theorem 3.4. So F is a weak free R-hypermodule on Y = l(X) over HSets. Let a ∈ A = 〈X〉. Acording to Proposition 3.1, suppose a ∈ m∑ i=1 rixi + n∑ i=1 nixi + k∑ i=1 ki(xi − xi) : ri ∈ R, xi ∈ X, m,n,k,ki ∈ N, ni ∈ Z}. Define f̄(z) = ∑m i=1 ril −1(exi ) or f̄(z) = ∑m i=1 rixi. The surjectivity of f̄ is clear. Y i // l−1∼= �� F ∃f̄ �� X i �� A (Note that since Y is a basis for F, we have z ∈ ∑ x∈X rxex for every z ∈ F where rx ∈ R as in the proof of Theorem 3.4.) � Int. J. Anal. Appl. 16 (6) (2018) 805 Now we state a new notion by using the fundamental module of an R-hypermodule. Definition 3.10. An R-hypermodule F is called fundamental free if its fundamental module, F �∗ F , is free R γ∗ -module. Example 3.1. [13] Let (G, ·) be a group with |G| ≥ 4, and define a hyperaddition and a multiplication on R = G∪{0}, by: a + 0 = 0 + a = a for all a ∈ R; a + a = {a, 0} for all a ∈ G; a + b = b + a = G\{a,b} for all a,b ∈ G : a 6= b; a� 0 = 0 �a = 0 for all a ∈ R; a� b = a · b for all a,b ∈ G. Then (R, +,�) is a hyperring. Clearly every hyperring R is an R-hypermodule and �∗R = γ ∗. On the other hand, for every x,y ∈ R we have xγ0γy, since x + x = {x, 0} and y + y = {y, 0}. So xγ∗y, and indeed we have only one equivalence class. Hence R′ = R γ∗ is the trivial ring 0 = {0R′}. Clearly R′ = Rγ∗ is a free R′-module. Thus R is fundamental free as an R-hypermodule. Remark 3.5. Note that in 3.1, every R-hypermodule is fundamental free, since every R′-module is free. Indeed, every module over the trivial ring R′ = 0 is free. In general, we state the following proposition: Proposition 3.3. For every hyperring R with trivial fundamental ring, all objects of RHmod (or Rhmod) are fundamental free. Proof. The proof is clear, since every R γ∗ -module is free. � Definition 3.11. An R-homomorphism f ∈ HomR(A,B) is �∗-inverse of g ∈ HomR(B,A) if �∗B((f ◦ g)(b)) = �∗B(b) and � ∗ A((g ◦ f)(a)) = � ∗ A(a) for all (a,b) ∈ A × B, or equivalently [f ◦ g] = [idB] and [g◦f] = [idA]. In this case, we say f is an �∗-isomorphism and A is �∗-isomorphic to B denoted by A �∗∼= B. Theorem 3.6. If F and F ′ are two �∗-free R-hypermodules on the sets X and X′ over HSets, respectively, and |X| = |X′|, then F �∗∼= F ′. Int. J. Anal. Appl. 16 (6) (2018) 806 Proof. Since |X| = |X′|, we have a bijection h : X −→ X′ in Sets. Now consider the inclusion i′ : X′ −→ F ′ and set f = i′ ◦ h as a morphism in HSets. Since F is �∗-free on X over HSets, we have an Rmv- homomorphism f̄ such that f̄ ◦ i = f as the diagram X i // h �� F f̄ �� X′ i′ // F ′ commutes. Also, consider the inclusion i : X −→ F and set g = i◦h−1 as a morphism in HSets. Since F ′ is �∗-free on X′ over HSets, we have an Rmv-homomorphism ḡ such that ḡ ◦ i′ = g as the diagram X′ i′ // h−1 �� F ′ ḡ �� X i // F commutes. Thus we have the following commutative diagram: X i // h �� F f̄ �� X′ i′ // h−1 �� F ′ ḡ �� X i // F So ḡ ◦ f̄ ◦ i = i◦h−1 ◦h = i◦ idX = i. On the other hand, idF ◦ i = i. Since F is �∗-free on X over HSets, we have [ḡ ◦ f̄] = [idF ]. Similarly, one can check that [f̄ ◦ ḡ] = [idF′]. Hence F �∗∼= F ′. � According to Theorem 3.2, we have the following result: Corollary 3.2. Let X ⊆ F and X′ ⊆ F ′ be two bases for R-hypermodules F and F ′, respectively. If F and F ′ are weak free on X and X′ over Sets, respectively, and |X| = |X′|, then F �∗∼= F ′. Lemma 3.1. Let X be a basis for an R-hypermodule. If every summation n∑ i=1 rixi for ri ∈ R and xi ∈ X, which xi’s are not necessarily distinct, is a singleton, then R is a ring and F is an R-module . Proof. Clearly, for all ri,rj ∈ R and xi,xj ∈ X, rixi + rjxj is a singleton. Also, (ri + rj)xi = rixi + rjxi is a singleton. Without loss of generality, assume x ∈ n∑ i=1 rixi and y ∈ n∑ i=1 r′ixi for ri,r ′ i ∈ R and xi ∈ X. Then we can write x + y ⊆ n∑ i=1 tixi in which ti = ri + r ′ i. According to the assumption, if we prove ti is a singleton, then x+y is a singleton and F is an R-module. Let r,s ∈ ti. Since tixi = rixi +r′ixi is a singleton and rxi,sxi ∈ tixi, we have rxi = sxi = tixi. Clearly, 0F ∈ rxi − sxi = (r − s)xi. On the other hand, Int. J. Anal. Appl. 16 (6) (2018) 807 0F ∈ 0R ∗xi. So 0R ∈ r −s. Hence r = s. Consequently, ti is a singleton. Now we prove R is ring. Let r,s ∈ R and t,t′ ∈ r + s. Clearly, txi, t ′xi ∈ (r + s)xi = rxi + sxi every xi ∈ X. Then txi = t′xi. Then 0F ∈ txi − t′xi = (t − t′)xi. On the other hand, 0F ∈ 0R ∗xi. So 0R ∈ t− t′. Hence t = t′. Consequently, r + s is a singleton. � Proposition 3.4. Let F be a free R-hypermodule on basis X ⊆ F such that for all r,s ∈ R and all x ∈ X, we have |rx + sx| = 1. Then R is a ring and F is a free R-module on X in the category RHmod. Proof. Note that every free R-hypermodule on X ⊆ F is a weak free R-hypermodule on X over Sets. Thus if i = idF |X, then X i // i �� F ∃!f~~ F i.e., f ◦ i = i. Since X is a basis, for an arbitrary x ∈ F, we have f(x) = ∑ xi∈X rxi xi in which every r x i ∈ R depends on x (by Equation 3.1). Note that ∑ xi∈X rxi xi is the unique presentation of x by the elements of the basis X, i.e., x ∈ ∑ xi∈X rxi xi and rxi ’s are unique. On the other hand, X i // i �� F idF~~ F and indeed, idF ◦i = i. So f = idF and thus f(x) = idF (x). This implies ∑ xi∈X rxi xi = x. Consequently, every unique presentation of every element of F is a singleton. According to the assumption, every summation n∑ i=1 rixi, which xi’s are not necessarily distinct, is a singleton. Thus the result is followed by Lemma 3.1. � References [1] R. Ameri, On categories of hypergroups and hypermodules, J. Discrete Math. Sci. Cryptogr. 6(2-3) (2003), 121–132. [2] S. Awodey, Category theory, Second ed., Oxford University Press, Inc. New York, 2010. [3] P. Corsini, Prolegomena of Hypergroup Theory, Second ed., Aviani Editore, Tricesimo, 1993. [4] P. Corsini and V. Leoreanu-Fotea, Applications of Hyperstructure Theory, Advances in Mathematics, Kluwer Academic Publication, Dordrecht, 2003. [5] B. Davvaz, A brief survey of the theory of Hv-structures, 8 th AHA, Greece, Spanidis (2003), 39–70. [6] B. Davvaz, Polygroup Theory and Related Systems, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013. [7] B. Davvaz and V. Leoreanu-Fotea, Hyperring Theory and Applications, International Academic Press, Palm Harbor, USA, 2007. [8] H. Herrlich and G. E. Strecker, Category theory, Vol. 2. Boston: Allyn and Bacon, 1973. [9] M. Krasner, A class of hyperrings and hyperfelds, Internat. J. Math. Math. Sci. 6 (1983), 307–311. Int. J. Anal. Appl. 16 (6) (2018) 808 [10] Ch G. Massouros, Free and cyclic hypermodules, Ann. Math. Pura Appl. 150 (1) (1988), 153–166. [11] F. Marty, Sur uni generalization de la notion de group, in: 8th Congress Math. Scandenaves, Stockholm, (1934), 45–49. [12] S. Sh. Mousavi and M. Jafarpour, On Free and Weak Free (Semi) Hypergroups, Algebra Colloq. 18 (2011), 873–880. [13] A. Nakassis, Expository and survey article of recent results in hyperring and hyperfield theory, Internat. J. Math. Math. Sci. 11 (1988) 209–220. [14] H. Shojaei and R. Ameri, Some results on categories of Krasner hypermodules, J. Fundam. Appl. Sci. 8 (3S) (2016), 2298–2306. [15] H. Shojaei, R. Ameri and S. Hoskova-Mayerova, On properties of various morphisms in the categories of general Krasner hypermodules, Italian J. Pure Appl. Math. 39 (2018), 475-484. [16] T. Vougiouklis, The fundamental relation in hyperrings, The general hyperfeld, 4th AHA, Xanthi 1990, World Scientifc (1991), 203–211. [17] T. Vougiouklis, Hyperstructures and Their Representations, Hadronic Press, Inc., 115, Palm Harber, USA, 1994. 1. Introduction 2. Preliminaries 3. Freeness of hypermodules References