Ratio Mathematica Vol. 32, 2017, pp. 45–62 ISSN: 1592-7415 eISSN: 2282-8214 Soft Γ- Modules Serkan Onar1, Sanem Yavuz2,Bayram Ali Ersoy3 1Yildiz Technical University, Department of Mathematics, Istanbul, Turkey serkan10ar@gmail.com 2Yildiz Technical University, Department of Mathematics, Istanbul, Turkey ssanemy@gmail.com 3Yildiz Technical University, Department of Mathematics, Istanbul, Turkey ersoya@gmail.com Received on: 14-04-2017. Accepted on: 28-06-2017. Published on: 30-06-2017 doi:10.23755/rm.v32i0.325 c©Serkan Onar et al. Abstract In this paper, the definitions of soft Γ -module, soft Γ - module ho- momorphism and soft Γ -exactness are introduced with the aid of the con- cept of soft set theory introduced by Molodtsov. In the meantime, some of their properties and structural characteristics are investigated and discussed. Thereafter, several illustrative examples are given. Keywords: soft set; soft module; Γ- ring; Γ -module; soft Γ- module; soft Γ -module homomorphism; soft Γ -module isomorphism; soft Γ -exactness. 2010 AMS subject classifications: 03E72; 08A72. 45 S. Onar, S. Yavuz and B. A. Ersoy 1 Introduction In the real world, there are some various uncertainties but classical mathemat- ical tools is not convenient for modeling these. Uncertain and unclear data which are contained by economy, engineering, environmental science, social science, medical science, business administration and many other fields are common. Al- though many diverse theories such as probability theory, soft set theory, intuition- istic fuzzy soft set theory and rough set theory are known and these present ad- vantageous mathematical approaches for modeling of uncertainties, each of these theories have their inherent diffuculties. In 1999, Molodstov [1] developed soft set theory which is considered a math- ematical tool for working with uncertainties. Since the emergence of soft set theory attracts attention and especially recently works on the soft set theory is progressing rapidly. Maji et al. [2] described some operations on soft sets and these operations are used soft sets of decision making problems. Chen et al. [3] offered a new definition for decrease of parametrerization on soft sets. They made comparasion between this definition and concept of restriction of property in the rough set theory. In theory, Maji et al. [4] worked various operator on soft set. Kong et al. [5] developed definition of parametrerization reduction on soft set. Zou and Xiao suggested some approach of data analysis in case of insufficent information on soft set. Jiang et al. presented a unique approach of the semantic decision making by means of ontological thinking and ontology-based soft sets. Besides studies on classic module theory have continued and interesting re- sults have been discovered recently. Macias Diaz et al. [6] studied on modules which are isomorphic to relatively divisible or pure submodules of each other. Abuhlail et al. [7] presented on topological lattices and their applications to mod- ule theory. On the other hand, Ameri et al. [8] investigated gamma module and Davvaz et al. [9] studied tensor product of gamma modules. As for soft module theory, Sun et al. [10] presented the notion of soft set and soft module. Xiang [11] worked soft module theory. T.Shah et al. [12] defined the notion of primary decomposition in a soft ring and soft module, and derived some related properties. Erami et al. [13] gave the concept of a soft MV- module and soft MV- submodule. In these days, there are some studies reletad with soft sets. Ali et al. [14] investigated some new operations in soft set theory and Pei et al. [15] studied from soft sets to information systems. Xiao et al. [16] presented research on synthetically evaluating method for business competitive capacity based on soft set. Aktaş et al. [17] showed soft sets and soft groups and Acar et al. [18] also showed soft sets and soft rings. The main purpose of this paper is to deal with algebraic structure of Γ− mod- ule by applying soft set theory. The concept of soft Γ− module is introduced, their characterization and algebraic properties are investigated by giving some several 46 Soft Γ- Modules examples. In addition to this, soft Γ− homomorphism , soft Γ− isomorphism and their properties are introduced. After all, we make inferences that images of soft Γ− homomorphisms and inverse images of soft Γ− homomorphisms are soft Γ− homomorphisms. Furthermore soft Γ− exactness is investigated and illustrated with a related example. 2 Preliminaries In this section, preliminary informations will be required to soft Γ− modules. First of all we give basic concepts of soft set theory. Definition 2.1. [18] Let X denotes an initial universe set and E is a set of pa- rameters. The power set of X is denoted by P (X). A pair of (F,E) is called a soft set over X if and only if F is a mapping from E into the set of all subsets of X, i.e, F : E → P (X). Definition 2.2. [18] Let (F,A) and (G,B) be two soft sets over a common uni- verse X. i) If A⊆̃ B and F (a)⊆̃ G (a) for all a ∈ A then we say that (F,A) is a soft subset of (G,B), denoted by (F,A) ⊆̃(G,B). ii) If (F,A) is a soft subset of (G,B) and (G,B) is a soft subset of (F,A), then we say that (F,A) is a soft equal to (G,B), denoted by (F,A) =̃ (G,B) . Example 2.1. Let X = M2(Z3) denotes an initial universe set, i.e, 2 × 2 ma- trices with Z3 terms and E = { [ 0 0 0 0 ] , [ 1 0 0 1 ] }is a set of parameters. Then F : E → P(X) where F( [ 0 0 0 0 ] ) = { [ 0 1 1 1 ] , [ 2 1 0 2 ] },F( [ 1 0 0 1 ] ) = { [ 2 0 0 2 ] }. Clearly,(F,E) is called a soft set over X. Definition 2.3. [18] Let (F,A) and (G,B) be two soft sets over a common uni- verse X. The intersection of (F,A) and (G,B) is defined as the soft set (H,C) satisfying the following conditions: i) C = A∩B. ii) For all c ∈ C, H (c) = F (c) or G (c) . In this case, we write (F,A)∩̃(G,B) = (H,C) . Definition 2.4. [18] Let (F,A) and (G,B) be two soft sets over a common uni- verse X. The union of (F,A) and (G,B) is defined as the soft set (H,C) satisfying the following conditions: i) C = A∪B. 47 S. Onar, S. Yavuz and B. A. Ersoy ii) For all c ∈ C, H (c) =   F(c) if c ∈ A−B, G(c) if c ∈ B −A, F(c) ∪G(c) if c ∈ A∩B.   This is denoted by (F,A)∪̃(G,B) = (H,C) . Definition 2.5. [18] If (F,A) and (G,B) are two soft sets over a common uni- verse X, then (F,A) AND (G,B) denoted by (F,A)∧̃(G,B) is defined as (F,A)∧̃(G,B) = (H,C), where C = A × B and H (x,y) = F (x) ∩ G (y), for all (x,y) ∈ C. Definition 2.6. Let {(Fi,Ai) : i ∈ I} be a non- empty family soft sets. The ∧−intersection of a non-empty family soft sets is defined by (ψ,Y ) = ∧̃i∈I (Fi,Ai) where (ψ,Y ) is a soft set, Y = ∏ i∈I Ai and ψ(y) = ∩i∈IFi(y) for every y = (yi)i∈I ∈ Y. Definition 2.7. [18] If (F,A) and (G,B) are two soft sets over a common universe X, then (F,A) OR (G,B) denoted by (F,A)∨̃(G,B) is defined as (F,A)∨̃(G,B) = (H,C), where C = A×B and H (x,y) = F (x) ∪G (y), for all (x,y) ∈ C. Definition 2.8. Let {(Fi,Ai) : i ∈ I} be a non- empty family soft sets. The ∨−union of a non-empty family soft sets is defined by (ψ,Y ) = ∨̃i∈I (Fi,Ai) where (ψ,Y ) is a soft set, Y = ∏ i∈I Ai and ψ(y) = ∪i∈IFi(y) for every y = (yi)i∈I ∈ Y. On the other hand we will introduce modules and soft modules, then we will study some properties and theories of soft modules such as trivial soft module, whole soft module, the concepts of soft submodule and soft module homomor- phisms. Definition 2.9. [10] Let R be a ring with identity. M is said to be a left R- module if left scalar multiplication λ : R × M → M via (a,x) 7→ ax satisfying the axioms ∀r,r1,r2, 1 ∈ R; m,m1,m2 ∈ M : i) M is an abelian group, ii) r(m1 + m2) = rm1 + rm2, (r1 + r2)m = r1m + r2m, iii) (r1r2)m = r1(r2m), iv) 1m = m. Left R−module is denoted by RM or M for short. Similarly we can define right R- module and denote it by MR. 48 Soft Γ- Modules Example 2.2. Let R = M2(Z) and M = { [ a b ] |a,b ∈ Z}. Then M is module on R. Definition 2.10. [10] Let M be a left R- module, A be a any nonempty set and (F,A) is a soft set over M. (F,A) is said to be a soft module over M if and only if F (x) is submodule over M, for all x ∈ A. Definition 2.11. [10] Let (F,A) be a soft module over M then i) (F,A) is said to be a trivial soft module over M if F(x) = 0 for all x ∈ A,where 0 is zero element of M. ii) (F,A) is said to be an whole soft module over M if F(x) = M for all x ∈ A. Proposition 2.1. [10] Let (F,A) and (G,B) be two soft modules over M. 1) (F,A)∩̃(G,B) is a soft module over M. 2) (F,A)∪̃(G,B) is a soft module over M if A∩B = ∅. Definition 2.12. [10] If (F,A) and (G,B) be two soft modules over M, then (F,A) + (G,B) is defined as (H,A×B), where H (x,y) = F (x) + G (y) for all (x,y) ∈ A×B. Proposition 2.2. [10] Assume that (F,A) and (G,B) are two soft modules over M.Then (F,A) + (G,B) is soft module over M. Definition 2.13. [10] Suppose that (F,A) and (G,B) be two soft modules over M and N respectively. Then (F,A)×(G,B) = (H,A×B) is defined as H(x,y) = F(x) ×G(y) for all (x,y) ∈ A×B. Proposition 2.3. [10] Let (F,A) and (G,B) be two soft modules over M and N respectively. Then (F,A) × (G,B) is soft module over M ×N. Definition 2.14. [10] Let (F,A) and (G,B) be two soft modules over M.Then (G,B) is soft submodule of (F,A) if i) B ⊂ A, ii) G(x) < F(x),∀ x ∈ B. This is denoted by (G,B)<̃(F,A). Proposition 2.4. [10] Let (F,A) and (G,B) be two soft modules over M.We say that (G,B) is soft submodule of (F,A) if G(x) ⊆ F(x),∀x ∈ A. Definition 2.15. [10] Assume that E = {e}, where e is unit of A.Then every soft module (F,A) over M at least have two soft modules (F,A) and (F,E) called trivial soft submodule. 49 S. Onar, S. Yavuz and B. A. Ersoy Proposition 2.5. [10] Let (F,A) and (G,B) are two soft modules over M and (G,B) is soft submodule of (F,A). If f : M → N is a homomorphism of module, then (f(F),A) and (f(G),B) are all soft modules over N and (f(G),B) is soft submodule of (f(F),A). Definition 2.16. [10] Let (F,A) and (G,B) be two soft modules over M and N respectively, f : M → N,g : A → B be two functions. Then we say that (f,g) is a soft homomorphism if the following conditions are satisfied: i) f : M → N is a homomorphism of module, ii) g : A → B is a mapping, iii) For all x ∈ A, f(F(x)) = G(g(x)). We say that (F,A) is a soft homomorphic to (G,B) which denoted by (F,A)−̃(G,B). In this definition, if f is an isomorphism from M to N and g is a one-to-one map- ping from A onto B, then we say that (F,A) is a soft isomorphism and that (F,A) is a soft isomorphic to (G,B), this is denoted by (F,A) =̃ (G,B) . Finally, we will define Γ- ring and Γ- module and their homomorphisms which are basic definitions for soft Γ- module. Definition 2.17. [8] Let R and Γ be additive abelian groups. Then we say that R is a Γ- ring if there exists a mapping: . : R× Γ ×R → R (r1,γ,r2) → r1γr2 such that for every a,b,c ∈ R and α,β ∈ Γ,the following hold: i) (a + b)αc = aαc + bαc, ii) a(α + β)c = aαc + aβc, iii) aα(b + c) = aαb + aαc, iv) (aαb)βc = aα(bβc). Definition 2.18. [8] A subset A of a Γ- ring R is said to be a right ideal of R if A is an additive subgroup of R and AΓR ⊆ A,where AΓR = {aαc| a ∈ A,α ∈ Γ,r ∈ R}. A left ideal of R is defined in a similar way. If A is both right and left ideal, we say that A is an ideal of R. Definition 2.19. [8] If R and S are Γ- rings, then a pair (θ,ϕ) of maps from R into S is called a homomorphism from R into S if i) θ(x + y) = θ(x) + θ(y), ii) ϕ is an isomorphism on Γ, iii) θ(xγy) = θ(x)ϕ(γ)θ(y). Definition 2.20. [8] Let R be a Γ- ring. A left Γ- module R is an additive abelian group M together with a mapping . : R × Γ × M → M such that for all m,m1,m2 ∈ M and γ,γ1,γ2 ∈ Γ,r,r1,r2 ∈ R the following hold: 50 Soft Γ- Modules i) rγ(m1 + m2) = rγm1 + rγm2, ii) (r1 + r2)γm = r1γm + r2γm, iii) r(γ1 + γ2)m = rγ1m + rγ2m, iv) r1γ1(r2γ2m) = (r1γ1r2)γ2m. A right Γ - module R is defined in analogous manner. Example 2.3. Let R = { [ k m ] |k,m ∈ Z2}, i.e, 1 × 2 matrices and Γ = { [ 0 0 ] , [ 1 0 ] } ∈ Z2, where Γis 2 × 1 matrices. Then we say that R is a Γ- ring. Similarly, R and Γ are same if we choose M = { [ 0 0 ] , [ 1 1 ] }, then M is Γ - module R. Definition 2.21. [8] Presume that (M, +) be an Γ - module R . A nonempty subset N of (M, +) is said to be a left Γ - submodule R of M if N is a subgroup of M and RΓN ⊆ N,where RΓN = {rγn |γ ∈ Γ,r ∈ R,n ∈ N}, that is for all n1,n2 ∈ N and for all γ ∈ Γ,r ∈ R; n1 −n2 ∈ N and rγn ∈ N. In this case we write N ≤ M. Example 2.4. In previous example, let N = { [ 0 0 ] }⊂ M and H : N → P(M) be a set valued function defined by H(a) = {b ∈ M|R(a,α,b) ⇔ aαb ∈ [ 0 0 ] } for all a ∈ N.H is clear that H( [ 0 0 ] ) = ( [ 0 0 ] ) is Γ - submodule R of M.. Definition 2.22. [8] Let M and N be arbitrary Γ - module R . A mapping f : M → N is a homomorphism of Γ - module R if for all x,y ∈ M and ∀r ∈ R,∀γ ∈ Γ we have i) f(x + y) = f(x) + f(y), ii) f(rγx) = rγf(x). A homomorphism f is monomorphism if f is one-to-one and f is epimorphism if f is onto. f is called isomorphism if f is both monomorphism and epimorphism. We denote the set of all RΓ- homomorphisms from M into N by HomRΓ (M,N) or shortly by HomRΓ (M,N). In particular M = N we denote Hom(M,M) by End(M). Definition 2.23. [18] Let M be a nonempty set and a Γ−module. The pair (F,A) is a soft set over M. The set Supp(F,A) = {x ∈ A : F(x) 6= ∅} is called a support of the soft set (F,A). The soft set (F,A) is non-null if Supp(F,A) 6= ∅. 3 Soft Γ- Modules In this section, firstly we will define soft Γ− modules, then we will give some operations on this modules.Throughout the section, M is a Γ−module. 51 S. Onar, S. Yavuz and B. A. Ersoy Definition 3.1. Let (F,A) be a non-null soft set over M. Then, (F,A) is said to be a soft Γ−module over M if F(a) is a Γ−submodule M such that F : A → P(M), (i.e. a → F(a)) for all a ∈ A,y ∈ Supp(F,A). Example 3.1. For consider the additively abelian groups Z6 = {0, 1, 2, 3, 4, 5} and Γ = {0, 2}. Let . : Z6 ×Γ×Z6 → Z6, (m1, Γ,m2) = m1Γm2. Hence Z6 is a Γ− module. Let A = Z6 and F : A → P(M) be a set valued function defined by f(0) = f(2) = f(4) = Z6, f(1) = f(3) = f(5) = {0, 3} are Γ−submodule of Z6. Hence (F,A, ) is a soft Γ−module over Z6. Example 3.2. Let M is a Γ− module and (F,A) be a soft set over M.F : A → P(M) is defined by F(x) = {y ∈ M| xαy = 0} for all x ∈ A,α ∈ Γ. It is clear that (F,A) is a soft Γ− module. Example 3.3. For consider the additively abelian groups M = R = { [ 0 0 ] , [ 1 0 ] , [ 0 1 ] , [ 1 1 ] }⊆ (Z2)1×2 and Γ = { [ 0 0 ] , [ 1 0 ] }⊆ (Z2)2×1 with addition defined as matrice addition. It is trivial that R is a Γ− ring. Also M is a Γ− module over R. Let N = { [ 0 0 ] } ⊆ M and H : N → P(M) be a set valued function defined by H(a) = {b ∈ M| R(a,α,b) ↔ aαb ∈ [ 0 0 ] ,∀α ∈ Γ} for all a ∈ N. It is clear that H( [ 0 0 ] ) = { [ 0 0 ] } are sub Γ− module of M. Hence (H,N) is soft Γ− module of M. Theorem 3.1. Let (F,A) and (G,B) are two soft Γ−modules over M. Then (F,A)∩̃(G,B) is a soft Γ−module over M if it is non-null. Proof. By definition, we have that (F,A)∩̃(G,B) = (H,C) where H(c) = F(x) ∩G(y) for all c ∈ C. We assume that (H,C) is a non-null soft set over M. If c ∈ Supp(H,C), then H(c) = F(x)∩G(y) 6= ∅. We know that (F,A) , (G,B) are both soft Γ−module over M, and so, the nonempty sets F(x) and G(y) are both Γ−submodule over M. Thus, H(c) is a Γ−submodule over M for all c ∈ Supp(H,C). In this position, (H,C) = (F,A)∩̃(G,B) is a soft Γ−module over M. 2 Theorem 3.2. Let (F,A) and (G,B) are two soft Γ−modules over M. Then (F,A)∪̃(G,B) is a soft Γ−module over M if A∩B = ∅. 52 Soft Γ- Modules Proof. By definition, we have that (F,A)∪̃(G,B) = (H,C) where H(c) = F(x) ∩ G(y) for all c ∈ C. Note first that (H,C) is a non-null owing to the fact that Supp(H,C) = Supp (F,A)∪̃(G,B) . Suppose that c ∈ Supp(H,C).Then H(c) 6= ∅ so we have F(x),G(y) 6= ∅. From the hypothesis A∩B = ∅, we follow that H(c) = F(x)∩G(y). On the other hand F(x)∩G(y) is a soft Γ−module over M, we conclude that (H,C) is a soft Γ−module over M for all c ∈ Supp(H,C). Consequently (F,A)∪̃(G,B) = (H,C) is a soft Γ−module over M. 2 On the other hand, union of two soft Γ− modules is not always soft Γ− mod- ule. We will explain this situation with following example. Example 3.4. Let M = Z6 = {0, 1, 2, 3, 4, 5} is a MΓ−module, Γ = {0, 1},A = Z2 = {0, 1} and B = Z3 = {0, 1, 2} such that F(0) = F(1) = {0, 2, 4},G(0) = G(1) = G(2) = {0, 3}A∩B = {0, 1}. If this condition is hold, then (F,A)∪̃(G,B) is not a soft Γ−module over M. Indeed, H(1) = {0, 2, 3, 4} /∈ P(M). Definition 3.2. If (F,A) and (G,B) are two soft Γ−modules over M, then (F,A) AND (G,B) denoted by (F,A)∧̃(G,B) is defined as (F,A)∧̃(G,B) = (H,C), where C = A×B and H (x,y) = F (x)∩̃G (y), for all (x,y) ∈ C. Theorem 3.3. Suppose that (F,A) and (G,B) are two soft Γ−modules over M. Then (F,A)∧̃(G,B) is soft Γ−module over M if it is non-null. Proof. Using definition, we have that (F,A)∧̃(G,B) = (H,C) where C = A×B and H (x,y) = F (x)∩̃G (y), for all (x,y) ∈ C. Then the hypothesis, (H,C) is a non-null soft set over M. Since (H,C) is a non-null, Supp (H,C) 6= ∅ and so, for (x,y) ∈ Supp (H,C) ,H (x,y) = F (x)∩̃G (y) 6= ∅. We assume that t1, t2 ∈ F (x)∩̃G (y) . In this position i) If t1, t2 ∈ F (x) = {y : R(x,y)} we have that xt1 ∈ A,xt2 ∈ A. This implies that x(t1 + t2) ∈ A. ii) If t1, t2 ∈ G (y) = {y1 : R(y,y1)} we have that yt1 ∈ B, yt2 ∈ B. This implies that y(t1 + t2) ∈ B. Hence F (x)∩̃G (y) is a Γ− submodule. By the definition of soft Γ− mod- ule, (F,A) and (G,B) are soft Γ−modules over M. F (x) ,G (y) are also Γ− submodule over M. Furthermore H (x,y) = F (x)∩̃G (y) is a Γ− submodule over M for all (x,y) ∈ (H,C) = (F,A)∧̃(G,B) . Hence (F,A)∧̃(G,B) is soft Γ−module over M. 2 Definition 3.3. If (F,A) and (G,B) are two soft Γ−modules over M, then (F,A) OR (G,B) denoted by (F,A)∨̃(G,B) is defined as (F,A)∨̃(G,B) = (H,C), where C = A×B and H (x,y) = F (x)∪̃G (y), for all (x,y) ∈ C. Theorem 3.4. Suppose that (F,A) and (G,B) are two soft Γ−modules over M. Then (F,A)∨̃(G,B) is soft Γ−module over M. 53 S. Onar, S. Yavuz and B. A. Ersoy Proof. Using definition, we have that (F,A)∨̃(G,B) = (H,C), where C = A × B and H (x,y) = F (x)∪̃G (y), for all (x,y) ∈ C. Assume that c ∈ Supp(H,C). Then H(c) 6= ∅ and so we have that F(x) 6= ∅,G(y) 6= ∅. By assumption, F (x)∪̃G (y) is a soft Γ− module of M for all c ∈ Supp(H,C). Consequently (F,A)∨̃(G,B) = (H,C) is a soft Γ−module over M. 2 Definition 3.4. Let (F,A) and (G,B) are two soft Γ−modules over M. Then (F,A) +̃ (G,B) = (H,A × B) is defined as H(x,y) = F(x) + G(y) for all (x,y) ∈ A×B. Theorem 3.5. Suppose that (F,A) and (G,B) are two soft Γ−modules over M. Then (F,A) +̃ (G,B) is soft Γ−module over M. Proof. By the definition we write (F,A) +̃ (G,B) = (H,A × B) and H(x,y) = F(x) + G(y) for all (x,y) ∈ A × B. Let (x,y) ∈ Supp(H,A × B).Then, H(x,y) 6= ∅ and so we have F(x) 6= ∅, G(y) 6= ∅. By taking into account, (F,A) and (G,B) are two soft Γ−modules over M, it follows that F(x) + G(y) is a soft Γ−module over M for all (x,y) ∈ Supp(H,A × B). Hence (F,A) +̃ (G,B) is soft Γ−module over M. 2 Definition 3.5. Let (F,A) and (G,B) are two soft Γ−modules over M. Then (F,A)×̃(G,B) = (H,A × B) is defined as H(x,y) = F(x) × G(y) for all (x,y) ∈ A×B. Theorem 3.6. Suppose that (F,A) and (G,B) are two soft Γ−modules over M. Then (F,A)×̃(G,B) is soft Γ−module over M. Proof. By the definition we write (F,A)×̃(G,B) = (H,A × B) and H(x,y) = F(x) × G(y) for all (x,y) ∈ A × B. Let (x,y) ∈ Supp(H,A× B). Then, H(x,y) 6= ∅ and so we have F(x) 6= ∅, G(y) 6= ∅. By taking into account, (F,A) and (G,B) are two soft Γ−modules over M, it follows that F(x)×G(y) is a soft Γ−module over M for all (x,y) ∈ Supp(H,A×B). Hence (F,A)×̃(G,B) is soft Γ−module over M. 2 Definition 3.6. Let (F,A) and (G,B) are two soft Γ−modules over M. Then (G,B) is called a soft Γ−submodule of (F,A) if i) B ⊆ A, ii) ∀b ∈ Supp(G,B),g(b) is a Γ−submodule of F (b) . This denoted by (G,B) ⊂ (F,A) . From the definition, it is easily deduced that if (G,B) is a soft Γ−submodule of (F,A) , then Supp(G,B) ⊂ Supp(F,A). Theorem 3.7. Let (F,A) and (G,B) be two soft Γ−modules over M and (F,A)⊆̃ (G,B) . Then (G,B) ⊂ (F,A) . 54 Soft Γ- Modules Proof. Straight forward. 2 Corolary 3.1. Let (F,A) be a soft Γ−module over M and {(Fi,Ai) : i ∈ I} be a nonempty family of soft Γ−submodules of (F,A) .Then, i) ∩̃i∈I (Fi,Ai) is a soft Γ−submodule of (F,A) if it is non-null. ii) ∪̃i∈I (Fi,Ai) is a soft Γ−submodule of (F,A) , if Ai∩Aj = ∅ for all i,j ∈ I and if it is non-null. iii) If Fi(ai) ⊆ Fj(aj) or Fj(aj) ⊆ Fi(ai) for all i,j ∈ I,ai ∈ Ai, then ∨̃i∈I (Fi,Ai) is a soft Γ−submodule of ∨̃i∈I (F,A) . iv) ∧̃i∈I (Fi,Ai) is a soft Γ−submodule of ∧̃i∈I (F,A) . v) The cartesian product of the family ∏̃ i∈I (Fi,Ai) is a soft Γ−submodule of∏̃ i∈I (F,A) . vi) ∑̃ i∈I (Fi,Ai) is a soft Γ−submodule of ∑̃ i∈I (F,A) . Proof. Similar to the proof of Theorems 3.5, 3.6, 3.9, 3.11, 3.13 and 3.15. 2 4 Soft Γ− Module Homomorphism In this section, firstly we will define trivial and whole soft Γ−modules over Γ−module M, homomorphism of Γ−modules and their properties. Moreover we will study soft Γ−module homomorphism and soft Γ−module isomorphism. Throughout the section, M is a Γ−module. Definition 4.1. Let (ρ,A) and (σ,B) be two soft Γ−modules over Γ−module M and Γ−module M1 respectively. Let f : M → M1 and g : A → B be two functions. The following conditions: i) f is an epimorphism of Γ−module, ii) g is a surjective mapping, iii) f(ρ(y)) = σ(ρ(y)) for all y ∈ A, were satisfied by the pair (f,g), then (f,g) is called soft Γ−module homo- morphism. If there exists a soft Γ−module homomorphism between (ρ,A) and (σ,B), we say that (ρ,A) is soft homomorphic to (σ,B), and is denoted by (ρ,A) ∼ (σ,B). If there exists a soft Γ−module isomorphism between (ρ,A) and (σ,B), we say that(ρ,A) is soft isomorphic to (σ,B), and is denoted by (ρ,A)−̃(σ,B). Definition 4.2. Let (F,A) be soft Γ−module over M. i) (F,A) is called the trivial soft Γ−module over M if F(a) = {0} for all a ∈ A. 55 S. Onar, S. Yavuz and B. A. Ersoy ii) (F,A) is called the whole soft Γ−module over M if F(a) = M for all a ∈ A. Definition 4.3. Let M and M1 be two Γ−modules and m : M → M1 a mapping of Γ−module. If (F,A) and (G,B) are soft sets over M and M1 respectively, then i) (m(F),A) is a soft set over M1 where m(F) : A → P(M1), m(F)(a) = m(F(a)) for all a ∈ A. ii) (m−1(G),B) is a soft set over M where m−1(G) : B → P(M),m−1(G)(b) = m−1(G(b)) for all b ∈ B. Corolary 4.1. Let m : M → M1 be an onto homomorphism of Γ−module. Then following statements can be given. i) (F,A) be soft Γ−module over M, then (m(F),A) is a soft Γ−module over Γ−module M1. ii) (G,B) be soft Γ−module over Γ−module M1, then (m−1(G),B) is a soft Γ−module over M. Proof. i) Since (F,A) is a soft Γ−module over M, it is clear that (m(F),A) is a non-null soft set over M1. For every y ∈ Supp(m(F),A) we have m(F)(y) = m(F(y)) 6= ∅. Hence m(F(y)) which is the onto homomorphic image of Γ−module F(y) is a Γ−module of M1 for all y ∈ Supp(F(m),A). That is (m(F),A) is a soft Γ−module over Γ−module M1. ii) It is easy to see that Supp(m−1(G),B) ⊆ Supp(G,B). By this way let y ∈ Supp(m−1(G),B).Then, G(y) 6= ∅. Hence m−1(G(y)) which is homomorphic inverse image of Γ−module G(y), is a soft Γ−module over M for all y ∈ B. 2 Theorem 4.1. Let m : M → M1 be a homomorphism of Γ−module and (F,A), (G,B) be two soft Γ−modules over Γ−module M and Γ−module M1 respec- tively. Then following statements can be given. i) If F(a) = ker (m) for all a ∈ A,then (m(F),A) is the trivial soft Γ−module over M1. ii) If m is onto and (F,A) is whole, then (m(F),A) is the whole soft Γ−module over M1. iii) If G(b) = m(M) for all b ∈ B,then (m−1(G),B) is the whole soft Γ−module over M. iv) If m is injective and (G,B) is trivial, then (m−1(G),B) is the trivial soft Γ−module over M. Proof. i) By using F(a) = ker (m) for all a ∈ A. Then m(F)(a) = m(F(a)) = {0M1} for all a ∈ A. Hence (m(F),A) is soft Γ−module over M1. 56 Soft Γ- Modules ii) Suppose that m is onto and (F,A) is whole. Then F(a) = M for all a ∈ A and so m(F)(a) = m(F(a)) = m(M) = M1 for all a ∈ A. Hence (m(F),A) is whole soft Γ−module over M1. iii) If we use hypothesis G(b) = m(M) for all b ∈ B, we can write m−1(G)(b) = m−1(G(b)) = m−1(m(M)) = M for all b ∈ B. It is clear that, (m−1(G),B) is the whole soft Γ−module over M. iv) Suppose that m is injective and (G,B) is trivial. Then, G(b) = {0} for all b ∈ B,so m−1(G)(b) = m−1(G(b)) = m−1({0}) = ker m = {0M} for all b ∈ B. Consequently, (m−1(G),B) is the trivial soft Γ−module over M. 2 Theorem 4.2. Let m : M → M1 be a homomorphism of Γ−module and (F,A), (G,B) be two soft Γ−modules over M. If (G,B) is soft Γ−submodule of (F,A), then (m(G),B) is soft Γ−submodule of (m(F),A). Proof. Suppose that y ∈ Supp (G,B). Then y ∈ Supp (F,A) .We know that B ⊆ A and G(y) is a Γ−submodule F(y) for all y ∈ Supp (G,B). From the expression hypothesis m is a homomorphism, m(G)(y) = m(G(y)) is a Γ−submodule of m(F)(y) = m(F(y)) and therefore (m(G),B) is soft Γ−submodule of (m(F),A). 2 Theorem 4.3. Let m : M → M1 be a homomorphism of Γ−module and (F,A), (G,B) be two soft Γ−modules over M. If (G,B) is soft Γ−submodule of (F,A), then (m−1(G),B) is soft Γ−submodule of (m−1(F),A). Proof. Let y ∈ Supp(m−1(G),B). B ⊆ A and G(y) is a Γ−submodule of F(y) for all y ∈ B. Since m is a homomorphism, m−1(G)(y) = m−1(G(y)) is a Γ−submodule of m−1(G(y)) = m(G)(y) for all y ∈ Supp(m−1(G),B). Hence (m−1(G),B) is soft Γ−submodule of (m−1(F),A). 2 5 Soft Γ− Exactness In this section, we will introduce maximal and minimal soft Γ−submodules. Then, we will investigate short exact and exact sequence of Γ−modules. Finally, we will explain soft Γ−exactness and some their basic theories. Throughout this section M is Γ−module. Definition 5.1. Let (F,A) and (G,B) be two soft Γ−modules over M and (G,B) be soft Γ−submodule of (F,A) . We say (G,B) is maximal soft Γ−submodule of (F,A) if G(x) is a maximal Γ−submodule of F(x) for all x ∈ B. We say (G,B) is minimal soft Γ−submodule of (F,A) if G(x) is a minimal Γ−submodule of F(x) for all x ∈ B. 57 S. Onar, S. Yavuz and B. A. Ersoy Proposition 5.1. Let (F,A) be a soft Γ−module over M. i) If {(Gi,Bi) |i ∈ I} is a nonempty family of maximal soft Γ−submodules of (F,A) , then ⋂ i∈I (Gi,Bi) is maximal soft Γ−submodule of (F,A) . ii) If {(Gi,Bi) |i ∈ I} is a nonempty family of minimal soft Γ−submodules of (F,A) , then ∑ i∈I (Gi,Bi) is minimal soft Γ−submodule of (F,A) . Proof. straight forward. 2 Corolary 5.1. Let (F,A) be a soft Γ−module over M and f : M → N be a homomorphism if F(x) = ker f for all x ∈ A, then (f(F),A) is the rivial soft Γ−module over N. Similarly, let (F,A) be an whole soft Γ−module over M and f : M → N be an epimorphism, then (f(F),A) is a whole soft Γ−module over N. Definition 5.2. The homomorphism sequence of Γ−modules ... → Mn−1 →fn−1 Mn →fn Mn+1 → ... is called exact sequence of Γ−modules if Imfn−1 = Kerfn for all n ∈ N and we call the exact sequence of Γ−modules form as 0 → M1 →f M →g M2 → 0 the short exact sequence of Γ−modules. Proposition 5.2. Let (F,A) be a trivial soft Γ−module over Γ−module M1 and (G,B) be a whole soft Γ−module over Γ−module M2 if 0 → M1 →f→ M →g→ M2 → 0 is a short exact sequence, then 0 → F(x) →f̃ M →g̃→ G(y) → 0 is a short exact sequence for all x ∈ A,y ∈ B. Proof. F(x) = 0,∀x ∈ A since (F,A) is a trivial soft Γ−module over Γ−module M1,so f̃ is a monomorphism. G(y) = M2,∀y ∈ B since (G,B) is a whole soft Γ−module over Γ−module M2.g : M → M2 is an epimorphism as 0 → M1 →f→ M →g→ M2 → 0 is a short exact sequence, so g̃ is an epimorphism. 2 Proposition 5.3. Let (F,A) be a trivial soft Γ−module over Γ−module M1 and (G,B) be a whole soft Γ−module over Γ−module M if 0 → M1 →f→ M →g→ M2 → 0 is a short exact sequence, then 0 → f(F)(x) →f̃ M →g̃ g(G)(y) → 0 is a short exact sequence for all x ∈ A,y ∈ B. Proof. F(x) = 0,∀x ∈ A since (F,A) is a trivial soft Γ−module over Γ−module M1.Kerf = 0, so Kerf = F(x),∀x ∈ A,consequently (f(F),A) is trivial soft Γ−module over M. (G,B) is a whole soft Γ−module over M and g : M → M2 is an epimorphism, so (g(G),B) is a whole soft Γ−module over M2, thus 0 → f(F)(x) →f̃ M →g̃ g(G)(y) → 0 is a short exact sequence for all x ∈ A,y ∈ B. 2 58 Soft Γ- Modules Definition 5.3. Let (F,A), (G,B) and (H,C) are three soft Γ−modules over Γ−modules M,N and K respectively. Then we say soft Γ− exactness at (G,B) , if the following conditions are satisfied: i) M →f1 N →f2 K is exact, ii) A →g1 B →g2 C is exact, iii) f1(F(x)) = G(g1(x)) for all x ∈ A, iv) f2(G(x)) = H(g2(x)) for all x ∈ B, which is denoted by (F,A) →(f1,g1) (G,B) →(f2,g2) (H,C). In this definition, if every (Fi,Ai), i ∈ I is soft Γ− exact, then we say that (Fi,Ai)i∈I is soft Γ− exact. Proposition 5.4. Let (F,A) and (G,B) are two soft Γ−modules over Γ−modules M and N respectively. If (F,A) →(f,g) (G,B) → 0 is soft Γ− exact, then (f,g) is soft Γ− homomorphism. In particular, if 0 → (F,A) →(f,g) (G,B) → 0 is soft Γ− exact, then (f,g) is soft Γ−isomorphism. Proof. Since (F,A) →(f,g) (G,B) → 0 is soft Γ− exact, we have M →f N → 0 and A →g B → 0 are exact. Thus f and g are epimorphisms, it is clear that (f,g) is homomorphism. If 0 → (F,A) →(f,g) (G,B) → 0 is soft Γ− exact, then 0 → M →f N → 0 and 0 → A →g B → 0 are exact. Thus f and g are isomorphisms, it is clear that (f,g) is soft Γ−isomorphism. 2 Definition 5.4. Let M = 0 and A = 0, then (F,A) = 0. We call (F,A) is a zero-soft Γ− module. Proposition 5.5. Let (F,A), (G,B) and (H,C) are three soft Γ−modules over Γ−modules M,N and K respectively. If (F,A) →(f1,g1) (G,B) →(f2,g2) (H,C) is soft Γ− exact with f1,g1 epimorphism and f2,g2 monomorphism, then (G,B) is a zero-soft Γ− module. Proof. Since (F,A) →(f1,g1) (G,B) →(f2,g2)→ (H,C) is soft Γ− exact with f1,g1 epimorphism and f2,g2 monomorphism, we have M →f1 N →f2 K and A →g1 B →g2 C, hence N = 0 and B = 0, it is clear that (G,B) is zero-soft Γ− module. 2 Theorem 5.1. Let (F,A) and (H,B) are two soft Γ−modules over Γ−modules M and N respectively. For any M ⊂ N,A ⊂ B and M ⊂ H(x) where x ∈ B. If (F,A) →(f,g) (H,B) is soft Γ−homomorphism, then 0 → (F,A) →(f,g) (H,B) →(f1,g1) (I,B/A) → 0 is soft Γ− exact, where I(x + A) = H(x)/M for all x ∈ B. 59 S. Onar, S. Yavuz and B. A. Ersoy Proof. We know that 0 → M →f N →f1 N/M → 0 and 0 → A →g B →g1 B/A → 0 are exact. It is clear that M is a Γ−submodule of N, so that N/M is a Γ−module and M is a Γ−submodule of H(x) and H(x)/M is always a Γ−submodule of N/M. This shows that (I,B/A) is a soft Γ−module over N/M. For all x ∈ B/A. Define f1 : N → N/M by f1(n) = n + M, for all n ∈ N. Meanwhile, we define g1 : B → B/A by g1(b) = b + A, for all b ∈ B. Therefore, it gives that f1(H(x)) = H(x) + M,I(g1(x)) = I(x + A) = H(x) + M for all x ∈ B, and hence f1(H(x)) = I(g1(x)).This implies 0 → (F,A) →(f,g) (H,B) →(f1,g1) (I,B/A) → 0 is soft Γ− exact. 2 Theorem 5.2. Let (F,A2), (G,A1) and (H,A) are three soft Γ−modules over Γ−modules M2,M1 and M respectively. If M1 and M2 are Γ−submodules of M with M2 ⊂ M1,A1 and A2 are Γ−submodules of A with A2 ⊂ A1, where M1 ⊂ H(x), for all x ∈ A and M2 ⊂ G(x) for all x ∈ A1. Then 0 → (I,A1/A2) →(f1,g1) (J,A/A1) →(f2,g2) (P,A/A1) → 0 is soft Γ− exact, where I(x + A2) = G(x)/M2, for all x ∈ A1, J(x + A2) = H(x)/M2, for all x ∈ A,P(x + A1) = H(x)/M1, for all x ∈ A. Proof. Since M1 and M2 are Γ−submodules of M with M2 ⊂ M1, we have a short exact sequence 0 → M1/M2 →f1 M/M2 →f2 M/M1 → 0. Since A1 and A2 are Γ−submodules of A with A2 ⊂ A1, there is a short exact sequence 0 → A1/A2 →g1 A/A2 →g2 A/A1 → 0. It is clear that M2 is a Γ−submodule of M1, so that M1/M2 is a Γ−module. It gives that G(x)/M2 is a Γ−module for all x ∈ A1 from M2 is a Γ−submodule of G(x). However G(x)/M2 is always a Γ−submodule of M1/M2. This shows that (I,A1/A2) is a soft Γ− module over M1/M2 for all x ∈ A1/A2. It is clear that (J,A/A2) and (P,A/A1) be a soft Γ− module over M/M2 and M/M1 respectively. Define f1 : M1/M2 → M/M2 by f1(m1 + M1) = m + M2, for all m1 ∈ M1. Meanwhile, we define g1 : A1/A2 → A/A2 by g1(a1 + A2) = a + A2, for all a1 ∈ A1. Therefore, we have f1(I(x)) = f1(G((x)/M2) = H(x) + M2,J(g1(x)) = J(x + A2) = H(x) + M2 for all x ∈ A1/A2, so f1(I(x)) = J(g1(x)) for all x ∈ A1/A2. Define f2 : M/M2 → M/M1 by f2(m + M2) = m + M1, for all m ∈ M. Let g2 : A/A2 → A/A1 be defined by g2(a + A2) = a + A1, for all a ∈ A. Also, we have f2(J(x)) = f2(H((x)/M2) = H(x) + M1 for all x ∈ A/A2, so f2(J(x)) = P(g2(x)) for all x ∈ A/A2. Hence 0 → (I,A1/A2) →(f1,g1) (J,A/A1) →(f2,g2) (P,A/A1) → 0 is soft Γ− exact. 2 60 Soft Γ- Modules Theorem 5.3. Let (Fi,Ai), i = 1, 2, 3, 4, 5 be a soft Γ−module over Γ−module Mi, i = 1, 2, 3, 4, 5 respectively. If 0 → (F1,A1) →(f1,g1) (F2,A2) →(f2,g2) (F3,A3) → 0 and 0 → (F3,A3) →(f3,g3) (F4,A4) →(f4,g4) (F5,A5) → 0 are soft Γ− exact. Then 0 → (F1,A1) →(f1,g1) (F2,A2) →(f3 f2,g3 g2) (F4,A4) →(f4,g4) (F5,A5) → 0 is soft Γ− exact. Proof. Since 0 → (F1,A1) →(f1,g1) (F2,A2) →(f2,g2) (F3,A3) → 0 and 0 → (F3,A3) →(f3,g3) (F4,A4) →(f4,g4) (F5,A5) → 0 are soft Γ− exact, we have 0 → M1 →f1 M2 →f2 M3 → 0 and 0 → M3 →f3 M4 →f4 M5 → 0 are exact. It is clear that 0 → M1 →f1 M2 →f3 f2 M4 →f4 M5 → 0 is exact. Since 0 → A1 →g1 A2 →g2 A3 → 0 and 0 → A3 →g3 A4 →g4 A5 → 0 are exact. It is clear that 0 → A1 →g1 A2 →g3 g2 A4 →g4 A5 → 0 is exact. Since f2(F2(x)) = F3(g2(x)) for all x ∈ A2 and f3(F3(x)) = F4(g3(x)) for all x ∈ A3. We have f3f2(F2(x)) = f3(F3(g2(x))) = F4(g3g2(x)) for all x ∈ A2. This implies 0 → (F1,A1) →(f1,g1) (F2,A2) →(f3 f2,g3 g2) (F4,A4) →(f4,g4) (F5,A5) → 0 is soft Γ− exactness. 2 6 Conclusion In this work the theoretical point of view of soft Γ− module is discussed. The work is focused on soft Γ− module, soft Γ− module homomorphism and soft Γ− exactness. By using these concepts, we studied the algebraic properties of soft sets in Γ− module structure. One could extend this work by studying other algebraic structures. References [1] D. Molodtsov, Soft set theory-First results, Computers Mathematics Appli- cations, 37(4-5)(1999), 19-31. [2] P.K. Maji, A.R. Roy, R. Biswas, An application of soft sets in a decision making problem, Computers Mathematics Applications, 44(2002), 1077- 1083. [3] D. Chen, E.C.C. Tsang, D.S. Yeung, X. Wang, The parameterization reduc- tion of soft sets and its applications, Computer Mathematics Applications, 49(2005), 757-763. [4] P.K. Maji, R. Biswas, A.R. Roy, Soft set theory, Computers Mathematics Applications, 45(2003), 555-562. 61 S. Onar, S. Yavuz and B. A. Ersoy [5] Z. Kong, L. Gao, L. Wang and S. Li, The normal Parameter Reduction of Soft Sets and Its Algoritm, Computers and Mathematics with Applications, 56 (1) 3029-3037. [6] J. E. Macias Diaz, On modules which are isomorphic to relatively divisible or pure submodules of each other, Quaestiones Mathematicae, 38 (6), 2015. [7] J. Abuhlail, C. Lomp, Topological lattices and their applications to modules, Journal of Algebra Its Applications, 15 (3), (2015). [8] R. Ameri, R. Sadeghi, Gamma modules, Ratio Mathematica 20, 127–147 (2010). [9] A. H. Rezaei, B. Davvaz, Tensor product of gamma modules, Article in Afrika Mathematica, 26 (7), 1601- 1608, (2015). [10] Q-M. Sun, Z-L. Zhang and J.Liu, Soft sets and modules ( In Guoyin Wang, Tian-rui Li, Jerzy W. Grzymala-Busse, Duoqian Miao, Andrzej Skowron,Yiyu 54 Yao Eds.) Rough sets and knowledge technology, RSKT, Proceedings,Springer, 403-409, (2008). [11] D.Xiang, Soft module theory, IEEE, 2013, 10th International Conference on Fuzzy Systems and Knowledge Discovery. [12] T.Shah and S.Medhit, Primary decomposition in a soft ring and soft module, Iranian Journal of Science & Technology, 38A3 (Special issue-Mathematics) 311-320, (2014). [13] A.Erami, A.Hasankhani and A.B.Saeid, MV-modules in view of soft set the- ory, Çankaya University Journal of Science and Engineering, Volume 13, No. 1 (2016) 001–015. [14] M. I. Ali, F. Feng, X. Liu, W.K. Min, and M. Shabir, On some new operations in soft set theory, Comput. Math. Appl., 57 (9), 1547-1553, (2009). [15] D. Pei and D. Miao, From soft sets to information systems, In proceedings of Granular Computing (Eds: X. Hu., Q. Liu, A. Skowron, T.Y.Lin, R.R. Yager, B. Zhang) IEEE, 2, 617-621. [16] Z. Xiao, Y. Li, B. Zhong and X.Yang, Research on synthetically evaluat- ing method for business competitive capacity based on soft set, Statical Re- search, 52-54. [17] H. Aktas, N. Cağman, Soft sets and soft groups, Inform. Sci. 177 (2007), 2726- 2735. [18] U. Acar, F. Koyuncu, B. Tanay, Soft sets and soft rings, Computers and Mathematics with Applications, 59 (11), 2010, 3458- 3463. 62