Application of soft sets to non-associative rings

1 Introduction

Many theories have been developed to deal with uncertainties, for example, theory of fuzzy sets [29], theory of intuitionistic fuzzy sets [4], theory of vague sets, theory of interval mathematics [5, 8] and theory of rough sets [15]. Though many techniques have been developed as a result of these theories, yet difficulties are seem to be there.

In order to model vagueness and uncertainties, D. Molodtsov introduced the concept of soft set theory and it has received much attention since its inception. In his paper [17], he presented the fundamental results of new theory and successfully applied it into several directions such as smoothness of functions, game theory, operations research, Riemann-integration, Perron integration, theory of probability etc. A soft set is a collection of approximate description of an object. He also showed how soft set theory is free from parametrization inadequacy syndrome of fuzzy set theory, rough set theory, probability theory and game theory. Soft systems provide a very general framework with the involvement of parameters. Research works on soft set theory and its applications in various fields are progressing rapidly in these years.

Maji [15, 16] presented an application of soft sets in decision making problems that is based on the reduction of parameters to keep the optimal choice objects. Chen [6] presented a new definition of soft set parametrization reduction and a comparison of it with attributes reduction in rough set theory. Pei and Miao [18] showed that soft sets are a class of special information systems. Kong [13] introduced the notion of normal parameter reduction of soft sets and its use to investigate the problem of sub-optimal choice and added parameter set in soft sets. In [3], some new operations are defined on soft sets and some old operations are redefined. A. Sezgin et al. [23], introduced the union soft subnear-rings and union soft ideals of a near-ring. Application of soft set theory in algebraic structures was introduced by Aktaş and Çağman [2]. They discussed the notion of soft groups and derived some basic properties. They also showed that soft groups extends the concept of fuzzy groups. Recently, X. Liu et al. [14], established some useful fuzzy isomorphism theorems of soft rings. They also introduced the concept of fuzzy ideals of soft rings. Jun [9, 10] investigated soft BCK/BCI-algebras and its application in ideal theory.

Kazim and Naseeruddin [12], have introduced the concept of left almost semigroups (LA-semigroups). A groupoid S is called an LA-semigroup if it satisfies the left invertive law: (ab) c = (cb) a for all a, b, c ∈ S. This structure is also known as Abel-Grassmann’s groupoid (abbreviated as AG-groupoid) [19, 20]. An AG-groupoid is the midway structure between a commutative semigroup and a groupoid. Later, in [11], Kamran extended the notion of LA-semigroup to left almost group (LA-group). A groupoid G is called a left almost group (LA-group), if there exists left identity e ∈ G that is ea = a for all a ∈ G, for a ∈ G there exists b ∈ G such that ba = e and left invertive law holds in G.

Left Almost Ring (LA-ring) is actually an off shoot of LA-semigroup and LA-group. It is a non-commutative and non-associative structure and gradually due to its peculiar characteristics it has been emerging as useful non-associative class which intuitively would have reasonable contribution to enhance non-associative ring theory. By an LA-ring, we mean a non-empty set R with at least two elements such that (R, +) is an LA-group,(R, ·) is an LA-semigroup, both left and right distributive laws hold. For example, from a commutative ring (R, + , ·) , we can always obtain an LA-ring (R, ⊕ , ·) by defining for all a, b ∈ R, a ⊕ b = b - a and a · b is same as in the ring.

Shah and Rehman [26], have discussed left almost ring (LA-ring) of finitely nonzero functions which is in fact a generalization of a commutative semigroup ring. Recently Shah and Rehman [27], discussed some properties of LA-rings through their ideals and intuitively ideal theory would be a gate way for investigating the application of fuzzy sets, intuitionistics fuzzy sets and soft sets in LA-rings. For example, T. Shah et al. [24], have applied the concept of intuitionistic fuzzy sets and established some useful results. In [22], some computational work through Mace4, has been done and some interesting characteristics of LA-rings have been explored. For some more study of LA-rings, we refer the readers to see ([21, 28]).

In this paper, by introducing soft LA-rings, we make a new approach to apply the Molodstov’s soft set theory to a class of non-associative rings. We do provide number of examples to illustrate the concepts of soft ideals, soft prime ideals and idealistic soft LA-rings. We organized the paper as follows: in Section 3, we define soft LA-ring, soft ideals, soft prime ideals and soft semi-prime ideals and establish various results by corresponding examples. In Section 4, in particular, we investigate the properties of idealistic soft LA-ring with respect to LA-ring homomorphism.

2 Preliminaries

In this section, we recall some basic notions relevant to soft sets and soft rings.

Definition 1. [17] Let U be an initial universe and E be a set of parameters. Let P (U) denotes the power set of U and A be a non-empty subset of E. A pair (F, A) is called a soft set over U, where F is a mapping given by F : A → P (U).

In other words, a soft set over U is a parametrized family of subsets of the universe U. For ɛ ∈ A, F (ɛ) may be considered as the set of ɛ-approximate elements of the soft set (F, A). Clearly, a soft set is not a set.

Definition 2. [16] For two soft sets (F, A) and (G, B) over a common universe U, we say that (F, A) is a soft subset of (G, B) if

A ⊆ B.

We write ( F , A ) ⊂ ˜ ( G , B ). (F, A) is said to be a soft super set of (G, B), if (G, B) is a soft subset of (F, A). We denote it by ( F , A ) ⊃ ˜ ( G , B ).

Definition 3. [16] Two soft sets (F, A) and (G, B) over a common universe U are said to be soft equal if (F, A) is a soft subset of (G, B) and (G, B) is a soft subset of (F, A).

Definition 4. [16] A soft set (F, A) over U is said to be a NULL soft set denoted by Φ if for all ɛ ∈ A, F (ɛ) =∅ ( null set).

Definition 5. [1] Let (F, A) and (G, B) be soft sets over a common universe U. The bi-intersection of (F, A) and (G, B) is defined as the soft set (H, C) satisfying the following conditions:

C = A ∩ B .

In this case, we write ( F , A ) ⊓ ∼ ( G , B ) = ( H , C ) .

Definition 6. [7] Let (F _i , A _i ) _i∈I be a nonempty family of soft sets over a common universe U. The bi-intersection of these soft sets is defined to be the soft set (H, C) such that C = ∩ _i∈IA _i , and H (x) = ∩ _i∈IF _i for all x ∈ C. So here we write ⊓ i ∈ I ∼ ( F i , A i ) = ( H , C ) .

Definition 7. [1] Let (F, A) and (G, B) be soft sets over a common universe U. The union of (F, A) and (G, B) is defined as the soft set (H, C) satisfying the following conditions:

C = A ∪ B .

In this case, it is denoted by ( F , A ) ∪ ∼ ( G , B ) = ( H , C ) .

Definition 8. [1] Let (F _i , A _i ) _iɛI  be a nonempty family of soft sets over a common universe U . The union of these soft sets is defined as the soft set (H, C) satisfying the followings conditions:

C = ∪  _{i ɛI}  A _i  .

It is denoted by ∪ i ɛ I ∼ ( F i , A i ) = ( H , C ) .

Definition 9. [1] Let (F, A) and (G, B) be soft sets over a common universe U, 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 10. [1] Let (F, A) and (G, B) be soft sets over a common universe U, 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 11. [7] Let (F _i , A _i ) _i∈I be a nonempty family of soft sets over a common universe U. The AND-soft set ∧ i ∈ I ∼ ( F i , A i ) of these soft sets is defined to be the soft set (G, B) such that B = Π_i∈IA _i and G (x) = ∩ _i∈IF _i  (x _i ) for all x = (x _i ) _i∈I ∈ B .

Similarly, the OR-soft set ∨ i ∈ I ∼ ( F i , A i ) of these soft sets is defined to be the soft set (G, B) such that B = Π_i∈IA _i and G (x) = ∪ _i∈IF _i  (x _i ) for all x = (x _i ) _i∈I ∈ B.

Definition 12. [7] Let(F, A) be a soft set.The set supp (F, A) = {x ∈ A ∣ F (x) ≠ ∅} is called the support of the soft set (F, A). A soft set is said to be non null if its support is not equal to the empty set.

3 Soft left almost rings

Definition 13. Let (F, A) be a non null soft set over LA-ring R. Then (F, A) is called a soft left almost ring (LA-ring) over R, if F (x) is a subLA-ring of R for all x ∈ R .

Throughout this paper, R denotes an LA-ring. We illustrate this definition by the following example.

Definition 14. Let (F, A) and (G, B) be soft LA-rings over R. Then (G, B) is called soft subLA-ring of (F, A) if it satisfies the following conditions:

B ⊂ A

Theorem 1. Let (F, A) and (G, B) be soft LA-rings R. Then

( F , A ) ∧ ∼ ( G , B ) is a soft LA-ring over LA-ring R if it is non-null.

Proof 1. (1) By Definition 9, let ( F , A ) ∧ ∼ ( G , B ) = ( H , C ), where C = A × B and H (x, y) = F (x) ∩ G (y), for all (x, y) ∈ C . Since by hypothesis (H, C) is non-null soft set over R and (x, y)∈ supp(H, C) , so H (x, y) = F (x) ∩ G (y) ≠ ∅ . As both F (x) and G (y) are subLA-rings, so this implies that H (x, y) is also a subLA-ring of R. Hence ( H , C ) = ( F , A ) ∧ ∼ ( G , B ) is a soft LA-ring over R.

(2) By Definition 5, we have ( F , A ) ⊓ ∼ ( G , B ) = ( H , C ), where H (x) = F (x)∩ G (x) ≠ ∅ for some x ∈ A ∩ B . Since the nonempty sets F (x) and G (x) both are subLA-rings of R and hence F (x) ∩ G (x) is a subLA-ring of R. Consequently ( F , A ) ⊓ ∼ ( G , B ) = ( H , C ) is a soft LA-ring over R.

Theorem 2. Let (F, A) and (G, B) be soft LA-rings over R. Then the following assertations are true:

If G (x) ⊂ F (x), for all x ∈ B ⊂ A, then (G, B) is a soft subLA-ring of (F, A) .

Proof 2. (1) Straight forward from the definition of soft subLA-ring.

(2) Let ( F , A ) ⊓ ∼ ( G , B ) = ( H , C ) . Since A ∩ B ⊂ A and also H (x) = F (x) ∩ G (x) is a subLA-ring of F (x), thus (H, C) is a soft subLA-ring of (F, A) . Likewise it can be proved that (H, C) is a soft subLA-ring of (G, B) .

Example 3. By Using Example 2, we have two soft LA-rings over R and let ( F , A ) ⊓ ∼ ( G , B ) = ( H , C ), where C = A ∩ B = {0, 1, 2, 3, 4, 5, 6, 7, 8} ∩ {0, 3, 8} = B = {0, 3, 8}. For every x ∈ C, we have H (x) = A (x) ∩ B (x) = {0, 3, 8} which is subLA-ring of both F (x) = {y ∈ R ∣ x · y ∈ {0, 3, 8}} and G (x) = {y ∈ R ∣ x + y ∈ {0, 3, 8}} . Consequently ( F , A ) ⊓ ∼ ( G , B ) is a soft subLA-ring of both (F, A) and (G, B) .

Theorem 3. 10 Let (F _i , A _i )  _iɛI be a nonempty family of soft LA-rings over R. Then the following are true:

∧ i ∈ I ∼ ( F i , A i ) is a soft LA-ring over R if it is non-null.

Proof 3. (1) By Definition 11, Let ∧ i ∈ I ∼ ( F i , A i ) = ( G , B ), where B = Π_i∈IA _i and G (x) = ∩ _i∈IF _i  (x _i ) for all x = (x _i ) _i∈I ∈ B . By hypothesis the soft set (G, B) is non-null. If x = (x _i ) _i∈I∈ supp(G, B), then G (x) = ∩ _i∈IF _i  (x _i ) ≠ ∅ . Thus the nonempty set F _i  (x _i ) is a subLA-ring of R, since (F _i , A _i ) is a soft LA-ring over R for all i ∈ I. Hence G (x) is a subLA-ring of R for all x∈ supp(G, B) and so ∧ i ∈ I ∼ ( F i , A i ) = ( G , B ) is a soft LA-ring over R.

(2) By Definition 8, we have ∪ i ɛ I ∼ ( F i , A i ) = ( G , B ) . Since (G, B) is non-null soft set and supp(G, B) = ∪ _i∈I supp (F _i , A _i ) ≠ ∅ , so B = ∪ _i∈I A _i and for all x ∈ B, G (x) = ∪ _i∈I(x)F _i  (x), where I (x) = {i ∈ I ∣ x ∈ A _i } . Now let x∈ supp(G, B) then G (x) =∪ _i∈I(x)F _i  (x) ≠ ∅ and obviously F _{i 0}  (x)≠ ∅ for some i _o  ∈ I (x). But by the hypothesis, {A _i  ∣ i ∈ I} are pairwise disjoint. Hence above i _o is indeed unique and so G (x) coincides with F _{i 0}  (x). Moreover, since (F _{i 0} , A _{i 0} ) is a soft LA-ring over R, therefore it implies that the nonempty set F _{i 0}  (x) is a subLA-ring of R and so G (x) = F _{i 0}  (x) is a subLA-ring of R for all x∈ supp(G, B). Consequently ∪ i ɛ I ∼ ( F i , A i ) = ( G , B ) is a soft LA-ring over R.

(3) By Definition 6, let ⊓ i ∈ I ∼ ( F i , A i ) = ( G , B ), where B = ∩ _i∈IA _i and G (x) = ∩ _i∈IF _i for all x ∈ B . Since (G, B) is a non-null soft set and if x∈ supp(G, B), then G (x) =∩ _i∈IF _i  ≠ ∅. As intersection of any number of subLA-rings of an LA-ring is a subLA-ring, hence for all i ∈ I, the nonempty set F _i  (x) is a subLA-ring of R. Hence (F _i , A _i ) is a soft LA-ring over R and therefore G (x) is a subLA-ring of R for all x∈ supp(G, B). Thus it follows that ⊓ i ∈ I ∼ ( F i , A i ) = ( G , B ) is a soft LA-ring over R.

Definition 15. Let (F, A) be a soft LA-ring over R. A non-null soft set (G, I) over R is called soft ideal of (F, A), denoted as ( G , I ) ⊲ ∼ ( F , A ), if it satisfies the conditions:

I ⊂ A

Example 4. Let us reconsider the LA-ring as defined in the Cayley Table of Example 1. Let A = {0, 1, 2, 3, 4, 5, 6, 7} and I = {0, 3, 4, 7}. Define a mapping F : A ⟶ P (R) by F (x) = {y ∈ R ∣ x · y ∈ {0, 4}}. Then F (0) = F (1) = F (2) = F (3) = {0, 1, 2, 3, 4, 5, 6, 7} and F (4) = F (5) = F (6) = F (7) = {0, 3, 4, 7} . Obviously all these sets are subLA-rings of R and hence (F, A) is soft LA-ring over LA-ring R . On the other hand, consider the set valued function G : I → P (R) given by G (x) = {y ∈ R ∣ x + y ∈ {0, 3, 4, 7}}, then G (0) = G (3) = G (4) = G (7) = {0, 3, 4, 7} . It follows that G (0) ⊲ F (0) , G (3) ⊲ F (3) , G (4) ⊲ F (4) , G (7) ⊲ F (7) . Hence (G, I) is a soft ideal of soft LA-ring (F, A) .

Theorem 4. Let (F₁, I₁) and (F₂, I₂) be soft idealsof soft LA-ring (F, A) over R. Then ( F 1 , I 1 ) ⊓ ∼ ( F 2 , I 2 ) is a soft ideal of (F, A) if it is non-null.

Proof 4. Since ( F 1 , I 1 ) ⊲ ∼ ( F , A ) and ( F 2 , I 2 ) ⊲ ∼ ( F , A ), so Using Definition 5, we can write (F₁, I₁) ⊓ ∼ (F₂, I₂) = (H, I) , where I = I₁ ∩ I₂ and for all x ∈ I, H (x) = F₁ (x) ∩ F₂ (x) . Also it is obvious that I ⊂ A . Suppose that the soft set (H, I) is non-null. If x ∈ supp(H, I), then F₁ (x)∩ F₂ (x) ≠ ∅. Since ( F 1 , I 1 ) ⊲ ∼ ( F , A ) and ( F 2 , I 2 ) ⊲ ∼ ( F , A ) , so there is a liberty to say that the nonempty sets F₁ (x) and F₂ (x) both are ideals of F (x). It follows that H (x) is an ideal of F (x) for all x ∈ supp(H, I). Hence ( F 1 , I 1 ) ⊓ ∼ ( F 2 , I 2 ) = ( H , I ) is a soft ideal of (F, A) .

Theorem 5. Let (F₁, I₁) and (F₂, I₂) be soft ideals of soft LA-rings (F, A) and (G, B) over LA-ring R respectively. Then ( F 1 , I 1 ) ⊓ ∼ ( F 2 , I 2 ) is a soft ideal of ( F , A ) ⊓ ∼ ( G , B ) if it is non-null.

Proof 5. By Definition 5 we write ( F 1 , I 1 ) ⊓ ∼ ( F 2 , I 2 ) = ( H , I ) , where I = I₁ ∩ I₂ and for all x ∈ I, H (x) = F₁ (x) ∩ F₂ (x) . Similarly we can write ( F , A ) ⊓ ∼ ( G , B ) = ( K , C ) and C = A ∩ B, where K (x) = F (x) ∩ G (x) for all x ∈ C. Since I₁ ∩ I₂ is non-null, there exists an x∈ supp (H, I) such that H (x) = F₁ (x)∩ F₂ (x) ≠ ∅. As I₁ ∩ I₂ ⊂ A ∩ B, we need to show that H (x) is an ideal of LA ring K (x) for all x∈ supp(H, I) . Because of the fact that F₁ (x) ⊂ F (x) and F₂ (x) ⊂ G (x), it can be observed that F₁ (x) ∩ F₂ (x) ⊆ F (x) ∩ G (x) . Hence H (x) is a subLA-ring of R. Finally we show that r . a ∈ H (x) for all r ∈K (x) and for all a ∈ H (x) . Since for all x ∈ C, F₁ (x) is an ideal of F (x) and F₂ (x) is an ideal of G (x) . Therefore for r ∈ K (x) = F (x) ∩ G (x) and a ∈ H (x) = F₁ (x) ∩ F₂ (x), it is observed that r . a ∈ F₁ (x) and r . a ∈ F₂ (x) . Hence r . a ∈ H (x) . Similarly we can show that a . r ∈ H (x) .

Example 5. Let (F, A) and (G, B) be soft LA-rings over R as defined in Example 2, where A = {0, 1, 2, 3, 4, 5, 6, 7, 8} and B = {0, 1, 2, 3, 4, 5, 6, 7}. Let define F : A ⟶ P (R) by F (x) = {y ∈ R ∣ x · y ∈ {0, 3, 8}} and the mapping G : B ⟶ P (R) is defined by G (x) = {y ∈ R ∣ x · y ∈ {0, 4}}. Using Cayley Tables of Examples 1 and 2, we see that (F, A) and (G, B) become soft LA-rings over R . Let we take I₁ = {0, 3, 8} and I₂ = {0, 3, 4, 7} . If we define the set valued functions F₁ : I₁ ⟶ P (R) and F₂ : I₂ ⟶ P (R) by F₁ (x) = {y ∈ R ∣ 3x + y = 3} and F₂ (x) = {y ∈ R ∣ x + y ∈ {0, 3, 4, 7}} respectively, then F₁ (0) = F₁ (3) = F₁ (8) = {3} and F₂ (0) = F₂ (3) = F₂ (4) = F₂ (7) = {0, 3, 4, 7} and obviously these are ideals of F (x) and G (x) respectively. Also for all x ∈ I₁ ∩ I₂, F₁ (x) ∩ F₂ (x) = {3} ⊲ F (x) ∩ G (x). This implies ( F 1 , I 1 ) ⊓ ∼ ( F 2 , I 2 ) is a soft ideal of ( F , A ) ⊓ ∼ ( G , B ).

Theorem 6. Let (F₁, I₁) and (F₂, I₂) be soft ideals of soft LA-ring(F, A) overR. If I₁ and I₂ are disjoint, then( F 1 , I 1 ) ∪ ∼ ( F 2 , I 2 ) is soft ideal of (F, A) .

Proof 6. Assume that ( F 1 , I 1 ) ⊲ ∼ ( F , A ) and ( F 2 , I 2 ) ⊲ ∼ ( F , A ). According to Definition 7, we can write ( F 1 , I 1 ) ∪ ∼ ( F 2 , I 2 ) = ( H , I ), where I = I₁ ∪ I₂ and for every x ∈ I, H ( x ) = { F 1 ( x ) if x ∈ I 1 - I 2 , F 2 ( x ) if x ∈ I 2 - I 1 , F 1 ( x ) ∪ F 2 ( x ) if x ∈ I 1 ∩ I 2 .

Clearly we have I ⊂ A. Suppose I₁ and I₂ are disjoint. Then for every x∈ supp(H, I) , we know that either x ∈ I₁ - I₂ or x ∈ I₂ - I₁ . If x ∈ I₁ - I₂, then H (x) = F₁ (x)≠ ∅ is an ideal of F (x) since ( F 1 , I 1 ) ⊲ ∼ ( F , A ). Similarly if x ∈ I₂ - I₁, then H (x) = F₂ (x)≠ ∅ is an ideal of F (x) since ( F 2 , I 2 ) ⊲ ∼ ( F , A ). Thus H (x) ⊲ F (x) for all x∈ supp(H, I) . Hence ( F 1 , I 1 ) ∪ ∼ ( F 2 , I 2 ) = ( H , I ) is soft ideal of (F, A) .

Theorem 7. Let (F, A) be a soft LA-ring over R and (G _i , A _i ) _i∈I be a nonempty family of soft ideals of (F, A). Then the following are true:

If ∧ i ∈ I ∼ ( G i , A i ) is non-null, then ∧ i ∈ I ∼ ( G i , A i ) is a soft ideal of the soft LA-ring (F, A).

Proof 7. (1) Using Definition 11, we can write ∧ i ∈ I ∼ ( G i , A i ) = (H, C), where C = Π_i∈IA _i , and H (x) = ∩ _i∈IG _i  (x _i ) for all x = (x _i ) _i∈I ∈ C. It remains to show that the non-null soft set (H, C) is a soft ideal of the soft LA-ring (F, A). Since for all i ∈ I, A _i  ⊂ A and C ⊂ A. Moreover, if x = (x _i ) _i∈I∈ supp(H, C) , then H (x) =∩ _i∈IG _i  (x _i ) ≠ ∅. Since (G _i , A _i ) is a soft ideal of (F, A) for all i ∈ I, we deduce that the nonempty set G _i  (x _i ) is an ideal of F (x _i ) for all i ∈ I. It follows that H (x) = ∩ _i∈IG _i  (x _i ) is an ideal of F (x _i ) for all x = (x _i ) _i∈I∈ supp(H, C) . Hence we conclude that ∧ i ∈ I ∼ ( G i , A i ) = ( H , C ) is a soft ideal of the soft LA-ring (F, A).

The proofs of (2) and (3) are similar to those of the corresponding parts of Theorem 3.

Definition 16. Let (F, A) be a soft LA-ring over R. A non-null soft set (H, I) over R is called soft prime ideal of (F, A) denoted by (H, I) ⊲  ^p  (F, A) , if

I ⊂ A .

Remark 1. It can be noted that every soft prime ideal is soft ideal but converse is not true. We illustrate this fact in the following example.

Example 7. Let R = A = {0, 1, 2, 3, 4, 5, 6, 7} is an LA-ring, I = {0, 3, 4, 7} ⊂A. Now consider the set-valued function F : A ⟶ P (R) given by F (x) = {y ∈ R ∣ x · y ∈ {0, 4}} . Using Example 1, it is can be shown that (F, A) is a soft LA-ring over LA-ring R . Define a mapping H : I → P (R) by H (x) = {y ∈ R ∣ x + y ∈ {0, 3, 4, 7}}, then H (0) = H (3) = H (4) = H (7) = {0, 3, 4, 7} and obviously H (0) ⊲ F (0) , H (3) ⊲ F (3) , H (4) ⊲ F (4) , H (7) ⊲ F (7) . So (H, I) = {H (0) , H (3) , H (4) , H (7)} = {0, 3, 4, 7}. It is not hard to see that F (3) · F (3) ∈ (H, I) , but F (3) ∉ (H, I) . Also if we take F (0) · F (3) ∈ (H, I) , then neither F (3) ∉ (H, I) nor F (0) ∉ (H, I). Hence (H, I) is a soft ideal of (F, A) but not a soft prime ideal of (F, A) .

Definition 17. Let (F, A) be a soft LA- ring over R. A non-null soft set (H, I) over R is called a soft semi-prime ideal of (F, A) denoted as (H, I) ⊲  ^sp  (F, A) if

I ⊂ A .

Theorem 8. Let (P, I) and (Q, J) be soft prime ideals of soft LA-ring (F, A) over R. If I and J are disjoint, then ( P , I ) ∪ ∼ ( Q , J ) is soft prime ideal of (F, A) .

Proof 8. By Definition 7, ( P , I ) ∪ ∼ ( Q , J ) = ( K , C ), where C = I ∪ J and for every x ∈ C, K ( x ) = { P ( x ) if x ∈ I - J , Q ( x ) if x ∈ J - I , P ( x ) ∪ Q ( x ) if x ∈ I ∩ J .

Since (P, I) ⊲  ^p  (F, A) and (Q, J) ⊲  ^p  (F, A) , clearly C ⊂ A. Suppose that I and J are disjoint. Then for all x∈ supp(K, C) , we know that either x ∈ I - J or x ∈ J - I. If x ∈ I - J, then K (x) = P (x)≠ ∅ is a prime ideal of F (x) since (P, I) is a soft prime ideal of (F, A). Similarly if x ∈ J - I, then K (x) = Q (x)≠ ∅ is a prime ideal of F (x) since (Q, J) is a soft prime ideal of (F, A). Thus K (x) is a prime ideal of F (x) for all x∈ supp(K, C) and hence ( P , I ) ∪ ∼ ( Q , J ) = ( K , C ) is soft prime ideal of (F, A) .

Theorem 9. Let (F, A) be a soft LA-ring over R and (G _i , I _i ) _i∈I be a nonempty family of soft prime ideals of (F, A) . If {I _i  ∣ i ∈ I} are pairwise disjoint, then ∪ i ∈ I ∼ ( G i , I i ) is a soft prime ideal of soft LA-ring (F, A) .

Proof 9. Clear from the Theorem 8.

4 Idealistic soft LA-rings

T. Shah and I. Rehman [26], have defined the notion of LA-ring homomorphism. In this section we define idealistic soft LA-ring and using the notion of LA-ring homomorphism and soft set theoretic approach we establish some results to study some characteristic properties of idealistic soft LA-ring.

Definition 18. Let (F, A) be a non-null soft set over R. Then (F, A) is called an idealistic soft LA-ring over R, if F (x) is an ideal of R for all x∈ supp(F, A) .

Example 8. Consider Example 4, (F, A) is an idealistic soft LA-ring over R since F (x) is an ideal of R for all x ∈ A.

We know that every ideal of an LA-ring R is a subLA-ring of R, similarly every idealistic soft LA-ring over R is a soft LA-ring over R. Following example shows that the converse is not true in general.

Let A = R = {0, 1, 2, 3, 4, 5, 6, 7, 8}. Define a set-valued function F : A ⟶ P (R) by F (x) = {y ∈ R ∣ x · y ∈ {0, 3, 8}} . Then F (0) = F (1) = F (2) = F (4) = F (5) = F (7) = F (8) = {0, 3, 8} and F (3) = F (6) = {0, 1, 2, 3, 4, 5, 6, 7, 8} . It can be easily observed that all these sets are subLA-rings over R . Therefore it follows that (F, A) is soft LA-Ring over R. Although F (8) = {0, 3, 8} is a subLA-ring but it is not an ideal of R because the condition for right ideal is not satisfied. Therefore (F, A) is not an idealistic soft LA-ring over R.

Proposition 1. Let (F, A) be a soft set over R and let B ⊂ A. If (F, A) is an idealistic soft LA-ring over R, then so is (G, B) whenever it is non-null.

Proof 10. Straight forward so omitted.

In the following we provide an example which shows that the converse of Proposition 1 is not true ingeneral.

Example 10. Let (F, A) be the soft set as considered in Example 9. It is clear that (F, A) is not an idealistic soft LA-ring over R. But on the other hand if we take B = {0, 3, 8} ⊂ A and by considering F| _B  (x) : {y ∈ R ∣ 3x + y = 3} , then clearly F| _B  (0) = f| _B  (3) = F| _B  (8) = {3} is an ideal of R for all x∈ supp(F| _B , B) . Therefore (F| _B , B) is an idealistic soft LA-ring over R.

Theorem 10. Let (F, A) and (G, B) be idealistic soft LA-rings over R. Then ( F , A ) ⊓ ∼ ( G , B ) is an idealistic soft LA-ring over R if it is non-null.

Proof 11. By Definition 5, we can write ( F , A ) ⊓ ∼ ( G , B ) = ( H , C ), where C = A ∩ B and H (x) = F (x) ∩ G (x) for all x ∈ C . Let assume that (H, C) is a non-null soft set over R. If x∈ supp (H, C), then H (x) = F (x)∩ G (x) ≠ ∅. Clearly the non-empty sets F (x) and G (x) both are ideals of R and therefore it follows that H (x) is an ideal of R for all x∈ supp(H, C) . Hence ( F , A ) ⊓ ∼ ( G , B ) = (H, C) is an idealistic soft LA-ring over R.

Theorem 11. Let (F, A) and (G, B) be idealistic soft LA-rings over R. If A and B are disjoint, then ( F , A ) ∪ ∼ ( G , B ) is an idealistic soft LA-ring over R.

Proof 12. Using Definition 7, ( F , A ) ∪ ∼ ( G , B ) = ( H , C ) , where C = A ∩ B and for all x ∈ C, H ( x ) = { F ( x ) if x ∈ A - B , G ( x ) if x ∈ B - A , F ( x ) ∪ G ( x ) if x ∈ A ∩ B .

Now suppose that A∩ B = ∅ and so for every x∈ supp(H, C), either x ∈ A - B or x ∈ B - A. As (F, A) is an idealistic soft LA-ring over R, so if x ∈ A - B, then obviously H (x) = F (x) is an ideal of R . Likewise, if x ∈ B - A, then H (x) = G (x) is an ideal of R. Consequently, H (x) is an ideal of R, for every x∈ supp(H, C) . Hence ( F , A ) ∪ ∼ ( G , B ) = ( H , C ) is an idealistic soft LA-ring over R.

Remark 2. In Theorem 11, if A and B are not disjoint, then ( F , A ) ∪ ∼ ( G , B ) may not be an idealistic soft LA-ring over R. Indeed, let R = {0, 1, 2, 3, 4, 5, 6, 7}, A = {2, 7} and B = {2}. Consider the set valued function F : A ⟶ P (R) defined by F (x) = {y ∈ R ∣ x + y ∈ {2, 7}}. Using Example 6, we have F (2) = F (7) = {2, 7} ⊲ R . Hence (F, A) is an idealistic soft LA-ring over R. Define a set valued function G : B ⟶ P (R) by G (x) = {y ∈ R ∣ x + y ∈ {2, 6}}. Then G (2) = {2, 6} ⊲ R and it follows that (G, B) is also an idealistic soft LA-ring over R. But F (2) ∪ G (2) = {2, 6, 7} is not an ideal of R and hence ( F , A ) ∪ ∼ ( G , B ) is not an idealistic soft LA-ring over R.

Theorem 12. Let (F, A) and (G, B) be idealistic soft LA-rings over R. Then ( F , A ) ∧ ∼ ( G , B ) is an idealistic soft LA-ring over R if it is non-null.

Proof 13. Let ( F , A ) ∧ ∼ ( G , B ) = ( H , C ), where C = A × B and H (x, y) = F (x) ∩ G (y) for all (x, y) ∈ C . Since (H, C) is non-null soft set over R, so H (x, y) = F (x) ∩ G (y) ≠ ∅ . Also given that (F, A) and (G, B) are idealistic soft LA-rings over R, therefore it follows that F (x) and G (y) are ideals of R and hence H (x, y) is an ideal of R for all (x, y)∈ supp(H, C). Thus ( H , C ) = ( F , A ) ∧ ∼ ( G , B ) is an idealistic soft LA-ring over R.

Definition 19. An idealistic soft LA-ring (F, A) over R is said to be trivial if F (x) = {0} for all x ∈ A. An idealistic soft LA-ring (F, A) over R is said to be whole if F (x) = R for all x ∈ A .

Example 11. Let R = A = {0, 1, 2, 3, 4, 5, 6, 7} and consider a set valued function F : A ⟶ P (R) given by F (x) = {y ∈ R ∣ y + 3x = 0}. Using Example 1, for every x ∈ A we have F (x) = {0} ⊲ R and hence (F, A) is a trivial idealistic soft LA-ring over R . If we define a set valued function G : A ⟶ P (R) by G (x) = {y ∈ R ∣ x · y ∈ {0, 3, 4, 7}}, then G (x) = R ⊲ R for every x ∈ A . Hence (G, A) is a whole idealistic soft LA-ring over R.

Let (F, A) be a soft set over R and let be a mapping of LA-rings. Then we can define a soft set (f (F) , A) over R, where f (F) : A ⟶ P (R) is given by f (F) (x) = f (F (x)) for all x ∈ A. By definition, it is easy to see that supp(f (F) , A) = supp(f, A).

Proposition 2. Let be an epimorphism of LA-rings. If (F, A) is an idealistic soft LA-ring over R, then (f (F) , A) is an idealistic soft LA-ring over .

Proof 14. Since (F, A) is a non-null soft set over R, so obviously (f (F) , A) is a non-null soft set over We see that for all x∈ supp(f (F) , A), f (F) (x) = f (F (x)) ≠ ∅ . As the non-empty set F (x) is an ideal of R and f is an epimorphism, so f (F (x)) is an ideal of Therefore f (F (x)) is an ideal of for all x∈ supp(f (F) , A). Hence (f (F) , A) is an idealistic soft LA-ring over

Theorem 13. Let (F, A) be an idealistic soft LA-ring over R and be an epimorphism ofLA-rings.

If F (x) = ker (f) for all x ∈ A, then (f (F) , A) is the trivial idealistic soft LA-ring over R′.

Proof 15. (1) Assume that F (x) = ker (f) for all x ∈ A . Then f (F) (x) = f (F (x)) = {0_R′} for all x ∈ A and by Proposition 2 and Definition 19, it follows that (f (F) , A) is the trivial idealistic soft LA-ring over R′.

(2) Assume that (F, A) is whole. Then F (x) = R for all x ∈ A. Hence f (F) (x) = f (F (x)) = f (R) = R′ for all x ∈ A and by Proposition 2 and Definition 19 (f (F) , A) is the whole idealistic soft ring over R′.

5 Conclusion

Nowadays, mathematics is becoming more and more non-associative and it is a general prediction that in few years’ non-associativity will govern mathematics and applied sciences. The motivation behind the generalization comes from the fact that the study of soft sets were previously restricted to associative algebraic structures, but on the other hand there are many sets which possess non-associative binary operations and we can dig out examples which investigate a non-associative structure in the context of soft sets. In this paper we made an effort to lay the foundation of the theory of soft non-associative ring. We initiated a step to apply soft sets (in Molodtsov’s sense) by considering a set of parameters as a non-associative structure and studied several related properties. We introduced the notions of soft LA-rings, soft subLA-rings. By defining soft ideals, soft prime ideals and soft semi-prime ideals, we investigated various related properties and illustrated these notions by number of corresponding examples. To extend this work, one could study the characterization of soft LA-ring by the properties of soft bi-ideals, soft quasi-ideals, soft-interior ideals, especially interms of soft M-systems. In the end we must say that this paper is just a beginning of a new structure and we studied a few ideas only, it is necessary to carry out more theoretical research to establish a general framework for the practical application.