STAR PROJECTIVE AND STAR INJECTIVE H v-MODULES

In this paper meantime the check and the defining concepts of product and direct sum, star projective and star injective in Hv-modules, we introduce a generalization extra of some notions in homological algebra to prove the five lemma and star projective and star injective Theorems in Hv-modules. We determine the conditions equivalent to split sequences in Hv-modules and also some interesting results on these concepts are given.


Introduction
A couple (H, •) of a non-empty set H and a mapping on H × H into the family of non-empty subsets of H is called a hyperstructure (or hypergroupoid).A hypergroup is a hyperstructure (H, •) with associative law: (x • y) • z = x • (y • z) for every x, y, z ∈ H and the reproduction axiom is valid: x • H = H • x = H for every x ∈ H, i.e., for every x, y ∈ H there exist u, v ∈ H such that y ∈ x • u and y ∈ v • x.This concept is introduced by Marty in 1934 [11].If A and B are non-empty subsets of H, then A • B is given by Also, x • A is used for {x} • A and A • x for A • {x}.Hyperrings, hypermodules and other hyperstructures are defined and several books have been written till now [1,2,8,14].The concept of H v -structures as a larger class than the well known hyperstructures was introduced by Vougiouklis at fourth congress of AHA (Algebraic Hyperstructures and Applications) [15], where the axioms are replaced by the weak ones, that is instead of the equality on sets one has non-empty intersections.The basic definitions and results of H v -structures can be found in [4,5,6,7,9,10,12,13,14].The fundamental relations, weak equality, weak commutative, weak monic, weak epic, weak isomorphism, star homomorphism, star isomorph, direct product and direct sum, isomorph sequences and star projective and split sequences in H v -modules are defined and is proved some results in [3,14,16,17].Also, some famous lemmas such as five short lemma, Snake lemma, Shanuels lemma are derived in the context of H v -modules.The notions of M [−] and −[M ] functors are introduced in [17] and the authors investigated the exactness of them and other problems.
The notion of exact sequences is a fundamental concept and it has been widely used in many areas such as ring and module theory.Our aim in this paper meantime the defining concept star injective, introduce a generalization extra of some notions in homological algebra to prove the five lemma and theorems star projective and star injective in H v -modules.Determine the conditions to split a sequence (in H v -modules) and finally some interesting results are given.

Basic concepts
The hyperstructure (H, ) is a weak associative hyperstructure where " • " hyperoperation is weak distributive with respect to " + "; i.e. for every x, y, z ∈ R we have By using a certain type of equivalence relations we can connect hyperstructures to ordinary structures.
Let (R, +, •) be an H v -ring.Vougiouklis in [14] defined the relation γ * as the smallest equivalence relation on R such that the quotient set R/γ * = {γ * (r) | r ∈ R} is a ring.The γ * is called the fundamental equivalence relation on R and R/γ * is called the fundamental ring.Let us denote the set of all finite polynomials of elements of R, over N, by U. We define the relation γ as follows: xγy ⇔ {x, y} ⊆ u, for some u ∈ U.
Theorem 2.1.[14] The relation γ * is the transitive closure of the relation γ, and the addition and multiplication operations on R/γ * are defined as follows: Now, suppose that M is an H v -module over an H v -ring R. Vougiouklis in [13] defined the relation ε * as the smallest equivalence relation on M such that the quotient {M/ε * (x) | x ∈ M } is a module over the ring R/γ * .The relation ε * is called the fundamental equivalence relation on M and M/ε * is called the fundamental module.Let us denote ϑ the set of all expressions consisting of finite hyperoperations either on R and M or the external hyperoperation applied on finite sets of elements of R and M [13].We consider the relation ε on M as follows: Theorem 2.2.[13] The relation ε * is the transitive closure of the relation ε, and the addition and external product on M/ε * are defined as follows: The heart of an H v -module M over an H v -ring R is denoted by ω M and is defined by , where 0 is the unit element of the group (M/ε * , ⊕).One can prove that the unit element of the group (M/ε * , ⊕) is equal to ω M .By the definition of ω M we have ω ω M = Ker(φ : Let M 1 and M 2 be two H v -modules over an H v -ring R and ε * 1 , ε * 2 and ε * be the fundamental relations on M 1 , M 2 and M 1 × M 2 respectively, then (x 1 , x 2 )ε * (y 1 , y 2 ) if and only if x 1 ε * 1 y 1 and x 2 ε * 2 y 2 for all (x 1 , x 2 ), (y 1 , y 2 ) ∈ M 1 × M 2 [13,14].Weak equality (monic, epic), exact sequences and relative results in H vmodules are defined as follows [3]: Let M be an H v -module.The non-empty subsets X and Y of M are weakly equal if for every x ∈ X there exists y ∈ Y such that ε * M (x) = ε * M (y) and for every where Ker( It is easy to see that every one to one (onto) strong homomorphism is weak monic (weak epic), but the converse is not true necessarily.In fact the concept of weak monic (weak epic) is a generalization of the concept of one to one (onto). Let ) and denote by f The sequences According to [3] for every strong homomorphism f : Let M be an H v -module and Then, f is weak epic if and only if F is onto.Moreover, f is weak monic if and only if F is one to one.Finally, f is a weak-isomorphism if and only if F is an isomorphism

Product and direct sum in H v -Modules
In this section, we meantime the check concept product and direct sum, we introduce a generalization of some notions in homological algebra to prove the five lemma in H v -modules.Also, we determine the conditions equivalent to split the exact sequences in H v -modules and some interesting results on these concepts are given.
Proposition 3.1.Let f : M −→ N and g : N −→ M be strong homomorphisms of H v -modules such that gf = 1.Then, N is the direct sum of Im(f ) and Ker(f ).
By applying f on Eq. ( 1) we obtain g(m) = gf (m 1 ) = m 1 .hence m 1 ∈ ω N .Since f is a strong homomorphism, it follows that m = f (m 1 ) ∈ ω N .So, Im(f ) ∩ Ker(f ) ⊆ ω N .Now, for every m ∈ M we have: then the product and direct sum coincide and will be written Similarly, we define hyperoperations on i∈I M i which if 0 = {m i } ∈ i∈I M i , then only finitely many of the a i are nonzero, say a i l , a i2 , • • •, a ir .Proposition 3.2.Let {M i } be a non-empty collection of H v -modules.For every H v -module X and every collection of strong homomorphisms {ψ i : M i −→ X} there exists an unique strong homomorphism ψ : i∈I M i −→ X defined by ψ i = ψι i such that for every i ∈ I the following diagram is commutative.
then the canonical injections ι i and projections Π i satisfy ( 1)-( 3) as the readers may easily verify.Likewise if Example 1.Note first that for any H v -module A, there are unique strong hornomorphisrns ω A −→ A and A −→ ω A .If A and B are any H v -modules then the sequences are exact, where the i and Π are the canonical injections and projections respectively.Similarly, if C is a submodule of D, then the sequence is exact, where i is the inclusion map and p is the canonical epimorphism.

Proposition 3.4. (Five Lemma in Hv-modules) Let
be a commutative diagram of H v -modules and H v -homomorphisms over an H v -ring R with both rows exact.Then, (1) if α 1 is weak monic and α 2 , α 4 weak epic then α 3 is weak epic.
(2) The proof is similar to the proof of (1).
(3) The proof follows from (1) and (2).( an exact sequence of H v -modules and H v -homomorphisms.Then ϕ is weak epic if and only if ψ is weak monic.
be an exact sequence of H vmodules and H v -homomorphisms such that ϕ is weak epic and ψ is weak monic, then (2) Let ϕ be weak epic and x ∈ Ker ψ.We have Im ϕ be a commutative diagram of H v -modules and H v -homomorphisms over an H v -ring R with both rows exact.Then, there is the exact sequence Withal, if f is weak monic, then f is weak monic and if ψ is weak epic, then ψ is weak epic.
Proof.We define ϕ(n + Im α) Now, if f is weak monic, since f is scowl the mapping f on Ker α, hence f is weak monic.Let ψ be weak epic.Consider n ∈ N .Then, n + Im γ ∈ N / Im γ.Since ψ is weak epic, then there is n ∈ N such that ϕ(n) be a star commutative diagram of H v -modules and strong homomorphisms which rows horizontal and diagonal are exact.If α is weak monic and β weak epic, then For the converse, let Ker Proposition 3.8.Let M 1 , M 2 and M be three H v -modules and the sequence is exact.Then the following conditions are equivalent.
of star homomorphisms and with bottom row exact.Since P is star projective, it follows that there exists a star homomorphism h : P −→ B such that ε * (gh(p)) = ε * (1 P (p)) for all p ∈ P .Therefore, the short exact sequence The exact sequence (3) is split, hence there exists a star homomorphism ψ : P −→ T 1 such that hψ we define the mapping h : P i −→ A by h(x) w = ϕι i (x) for every x ∈ P i .Now, we have gh(x) Then gh w = f .Nothing that h is a star homomorphism.Therefore, P i is star projective.The converse is proved by a similar techniques and using the diagram If each P i is star projective, then for each i there exists a star homomorphism h i : P i −→ A such that gh i w = f ι i .By Theorem 3.2 there is a unique star homomorphism h : Verify that gh = f .Definition 4.4.An H v -module P is a star injective if for every diagram of star homomorphisms and H v -modules as follows  Then gh w = f .Therefore E i is star injective.The converse can be proved by a similar techniques and using the diagram If each E i is star injective, then for each i ∈ I there exists a star homomorphism h i : B −→ E i such that h i g w = Π i f .By Theorem 4.4 of [17], there is a unique star homomorphism h : B −→ i∈I E i with Π i h w = h i for every i.Hence h w = ι i h i .Then

w
= ϕ(n ) + Im β and ψ(n + Im β) w = ψ(n) + Im γ.Also we define f and g, scowl the mapping f and g on Ker α and Ker β respectively.It can be easily we seen that ϕ, ψ, f and g are well define also Im f w = Im f , Ker g w = Ker g, Im ϕ w = Im ϕ, Ker ψ w = Ker ψ.We have Im f w = Ker g and Im ϕ w = Ker ψ.Now, we defining the mapping d : Ker γ −→ N / Im α by d(m ) = n + Im α.We show that d is well define.If m " ∈ Ker γ, then γ(m " ) w = ω N " .Since g is weak epic, then there is m ∈ M such that g(m) w = m " .Thus γg(m) w = ω N " .We obtain ψβ(m) w = ω N " .Hence β(m) ∈ Ker ψ w = Im ϕ.Therefor, there is n ∈ N such that β(m) w = ϕ(n ).Since ϕ is weak monic, then n is unique.Thus, if m 1 = m 2 we obtain g(m 1 ) w = g(m 2 ).Then there is n 1 , n 2 ∈ N such that ϕ(n 1 ) w = β(m 1 ) and ϕ(n 2 ) w = β(m 2 ).Therefor n 1 + Im α w = n 2 + Im α.Then d is well define.It can be easily we seen that d is a H v -homomorphism.Now, we have Im d = {n = Im α | m " Ker γ} and Ker ϕ

w=
ι i f .We define the mapping h :B −→ E i by h(x ) w = Π i ϕ(x ) for every x ∈ B. Now for every x ∈ A we have hg(x) w = Π i ϕg(x) w = Π i ι i f (x) w = f (x).
T2of H v -modules and star homomorphism such that bottom row is exact, there is a star homomorphism ϕ : P −→ T 1 such that ε * (gϕ(p)) = ε * (f (p)) for all p ∈ P .Now, we consider the mapping h : T 1 −→ P by h(t) ∈ f −1 g(t) for every t ∈ T 1 .since f −1 and g are star homomorphisms.Hence h is star homomorphism.Now, we have the exact sequence 4.3.Let R be an H v -ring.A direct sum i∈I P i of H v -modules is star projective if and only if each P i is star projective.
Proposition 4.5.Let R be an H v -ring.Then, (1) If E is a star injective H v -module, then every short exact sequence According to what was said in case H v -modules star projective, since f −1 and g are star homomorphisms.Hence h is a star homomorphism.Now Coker h . is an exact sequence.By hypothesis, the above exact sequence is split.So, there exists a star homomorphism ϕ : T 2 −→ E such that ϕh Proposition 4.6.Let R be an H v -ring.A direct product i∈I E i of H v -modules is star injective if and only if each E i is star injective.
v -modules and star homomorphisms such that upper row exact.Since E is star injective, it follows that there is a star homomorphism h :A −→ E such that ε * (hg(e)) = ε * (1 E (e))) for all e ∈ E. Therefore, the short exact sequencew E / / E f / / A g / / B / /w E is split, by Theorem 3.8 and A w = B E. f / / A g / / B / / w E is split.We show that for every diagram f T 2 E of H v -modules and star homomorphism such that upper row is exact, there is a star homomorphism ϕ : T 2 −→ E such that ε * (ϕg(t 1 )) = ε * (f (t 1 )) for all t 1 ∈ T 1 .Now we defined the maping h : E −→ T 2 by h(e) ∈ gf −1 (e) for every e ∈ E. h / / T 2 / / Coker h / / w w = 1 E f w = f .Then ϕ(hf ) w = ψ(g) w = f .hence ϕg w = f .then E is star injective.i∈I E i is star injective, it follows that there exists a star homomorphism ϕ : B −→ i∈I E i such that ϕg