An investigation of soft radicals

1 Introduction

In many scientific studies using mathematics, the solution of uncertainties is of great importance. Since the modelling of uncertain data in many fields is very complex, it was difficult to successfully deal with them by classical methods. With the emergence of fuzzy set [17] and soft set theories [13], overcoming these problems have been a major field of study for mathematicians. A lot of work has been done on these theories both in applied and theoretical fields and it is continuing with an increasing speed.

Soft set theory has provided various solutions to the information systems and decision-making methods that are developed today. These solutions also increased the value of the theoretical studies. The new algebraic operations have been described by various researchers [3, 11]. These studies have facilitated the transfer of theoretical studies to practical areas as in the following studies: Maji et al. [12], Zou and Xiao [18] and some other authors as Atagun et al. [14], Shah and Medhit [16]. In addition, by defining the soft group [2] in 2007, the soft set theory has experienced rapid growth in algebraic structures. With the definition of the soft group, soft forms of rings, semi-groups, fields, modules and substructures of these structures have been defined as in the following studies: Atagun and Sezgin [5], Acar et al. [1], Shah and Medid [16], Ali et al. [4], Atagun and Aygun [6], Feng et al. [9] and Sezgin and Atagun [15].

Our aim is to be able to define the radicals of these ideals, taking into account all of the obtained soft ideal studies. First, by examining the defined soft ideal structures in the ring, we have defined soft radicals with different definitions in [1] and [5]. Then, by transferring the radical of an ideal to a soft ideal definition in [4], we have tried to obtain a soft radical in the semigroup. In this study, the main theorems which are obtained by considering the radical of an ideal for a ring and soft algebraic operations are included. This paper is illustrated by several examples.

2 Some basic definitions

Definition 2.1. A pair (Φ, Γ) is called a soft set over U, where Φ is a mapping Φ : Γ → P (U) [13]. In other words, a soft set over U is a parameterized family of subsets of the universe U. For x ∈ Γ, Φ (e) is considered as the set of e-elements of the soft set (Φ, Γ) or as the set of e-approximate elements of the soft set. For a soft set (Φ, Γ) , the set Supp (Φ, Γ) = {x ∈ Γ|Φ (x) ≠ Ø} .

Definition 2.2. Let Λ be a group, Γ be non-empty set and (Φ, Γ) be soft set over Λ. Then (Φ, Γ) is said to be a soft group over Λ if Φ (x) is a subgroup of Λ for all x ∈ Γ [2].

Definition 2.3. Let Λ be a ring and (Φ, Γ) be a non-null soft set over a ring Λ. Then (Φ, Γ) is called a soft ring over Λ if Φ (x) is a subring of Λ for all x ∈ Γ [1].

Definition 2.4. Let Γ be an ideal of ring Λ and let (Φ, Γ) be a soft set over Λ. If for all x, y ∈ Γ and r ∈ Λ,

Φ (x - y) ⊇ Φ (x) ∩ Φ (y) ,

Then (Φ, Γ) is called a soft ideal of Λ and denoted by (Φ, Γ) ⊴ Λ [5].

Definition 2.5. Let (Φ, B) be a soft ring over a ring Λ. A non-null soft set (γ, A) over Λ is called soft ideal of (Φ, B) which will be denoted by (γ, A) ⊴ (Φ, B) if it satisfies the following conditions:

A ⊂ B,

Definition 2.6. A soft set ( α , L ) over a semigroup S is called a soft semigroup if N ∨ ( S , L ) ≠ ( α , L ) ≠ Ø ∨ S and ( α , L ) ∘ ˆ ( α , L ) ⊆ ( α , L ) [4].

Definition 2.7. A soft set N ∨ ( S , L ) ≠ ( α , L ) ≠ Ø ∨ S over a semigroup S is called a soft left(right) ideal over S , L S ∨ ∘ ˆ ( α , L ) ⊆ ( α , L ) ( ( α , L ) ∘ ˆ L S ∨ ⊆ ( α , L ) ) , where L S ∨ is an absolute soft set over S . A soft set over S is a soft ideal if it is both a soft left and soft right ideal over S [4].

Definition 2.8. Let I be an ideal over commutative ring Λ. Then I = { x ∈ Λ : xn ∈ I } is said to be the radical of an ideal I for some positive integer. Equivalently, the radical of I is the pre-image of the ideal of nilpotent elements (called nil radical) in Λ / I [7].

3 Soft radicals

In this section we first define the soft radical for a ring. Then, we define the soft radical for a soft ring. After all, we transfer the soft radical definition to the soft ideal for a semigroup. From now on Λ will denote a commutative ring.

Definition 3.1. Let Λ be a ring, I be an ideal of Λ and ( α , I ) be a soft ideal over Λ. Then the soft set ( α , Λ ) is called an ideal-soft radical corresponding to ideal I , defined by α ( r ) = { α ( r n ) , r n ∈ I Ø , r n ∉ I , where α : Λ → ℘ ( Λ ) for all r ∈ Λ and some n ∈ ℕ . The ideal-soft radical ( α , Λ ) can be denoted by ( α , I ) .

Example 3.2. Let Λ = ( ℤ 6 , + , . ) be a ring and I = { 0 ¯ , 2 ¯ , 4 ¯ } be the ideal of Λ. Consider the function α : I → ℘ ( ℤ 6 ) defined by α ( 0 ¯ ) = ℤ 6 , α ( 2 ¯ ) = { 2 ¯ , 4 ¯ } and α ( 4 ¯ ) = { 2 ¯ , 4 ¯ } .

( α , I ) = { ( 0 ¯ , ℤ 6 ) , ( 2 ¯ , { 2 ¯ , 4 ¯ } ) , ( 4 ¯ , { 2 ¯ , 4 ¯ } ) } is a soft ideal of Λ. Since there exists some n ∈ ℕ such that 0 ⁿ , 2 n , 4 n ∈ I but 1 n , 3 n , 5 n ∉ I . Therefore, ( ( α , ℤ 6 ) = { ( 0 ¯ , ℤ 6 ) , ( 1 ¯ , Ø ) , ( 2 ¯ , { 2 ¯ , 4 ¯ } ) , ( 3 ¯ , Ø ) , ( 4 ¯ , { 2 ¯ , 4 ¯ } ) , ( 5 ¯ , Ø ) } is an ideal-soft radical over Λ .

Lemma 3.3. Let ( α , I ) be a soft ideal over a ring Λ. Then, α (rx - sy) ⊇ α (rx) ∩ α (sy) for all r, s ∈ Λ and x , y ∈ I .

Proof. Since I ⊴ Λ , there exist k , t ∈ I with rx = k and sy = t for all r, s ∈ Λ and x , y ∈ I . Hence, by Definition 2.4, α (k - t) ⊇ α (k) ∩ α (t) which completes the proof.

Theorem 3.4. Let Λ be a ring, I be an ideal of Λ and ( α , I ) be a soft set over Λ. If ( α , I ) is a soft radical over Λ, then ( α , I ) is a soft ideal of Λ and ( α , I ) ⊇ ( α , I ) .

Proof. Suppose Λ is a ring, I is an ideal of Λ and ( α , I ) is a soft set over Λ . By Definition 3.1, if x ∈ I then x n ∈ I therefore ( α , I ) ⊇ ( α , I ) for some n ∈ ℕ . If x m ∈ I and y n ∈ I for x, y ∈ Λ and for some m , n ∈ ℕ , then (x, α (x ^m )) and ( x , α ( y n ) ) ∈ ( α , I ) . Since ( x - y ) m + n = ( m + n 0 ) . x m + n - ( m + n 1 ) . x m + n - 1 . y + ( m + n 2 ) . x m + n - 2 . y n … - ( m + n m + n - 1 ) . x . y m + n - 1 + ( m + n m + n ) . y m + n , one can easily see that ( x - y ) m + n ∈ I . Since ( x - y ) m + n ∈ I , we have α ( ( x - y ) m + n ) = α ( ( m + n 0 ) . x m + n - ( m + n 1 ) . x m + n - 1 . y + ( m + n 2 ) . x m + n - 2 . y n … + ( m + n m + n - 1 ) . x . y m + n - 1 + ( m + n m + n ) y m + n ) = α ( ( … ) x m + ( … ) y n ) ⊇ α ( x m ) ∩ α ( y n ) .

By Lemma 3.3, we obtain that α ((x - y) ^m+n) ⊇ α (x ^m ) ∩ α (y ⁿ ). Therefore, we have α ( x - y ) ⊇ α ( x ) ∩ α ( y ) .

If x m ∉ I and y n ∈ I or x m ∉ I and y n ∉ I for x, y ∈ Λ and some m , n ∈ ℕ , then α (x ^m ) ∩ α (y ⁿ ) = Ø . Hence in this situations, it can be easily seen that α ( x - y ) ⊇ α ( x ) ∩ α ( y ) .

If x n ∈ I for x ∈ Λ and some n ∈ ℕ , then Λ is a commutative ring ( ( rx ) n ) = ( rx . rx . rx … rx ) = r n . x n ∈ I

Hence α (r ⁿ  . x ⁿ ) ⊇ α (x ⁿ ) and we have α ( rx ) ⊇ α ( x ) .

If x n ∉ I for x ∈ Λ and some n ∈ ℕ , then ( ( rx ) n ) = ( rx . rx . rx … rx ) = r n . x n ∉ I .

Therefore, α ( rx ) = Ø ⊇ Ø = α ( x ) .

Since Λ is a commutative ring, α ( xr ) ⊇ α ( x ) . Thus ( α , I ) is a soft ideal of Λ .

Theorem 3.5. Let ( α , I ) be a soft ideal of ringΛ. Then, there exists a soft ideal ( α , L ) such that ( α , I ) = ( α , L ) ⇔ ( α , I ) = ( α , I ) .

Proof. Suppose that there exists a soft ideal ( α , I ) such that ( α , I ) = ( α , L ) . If for some n ∈ ℕ and x _k  ∈ Λ, ( x k ) n ∈ L , then we have ( x k ) n ∈ L ⇔ ( x k α ( ( x k ) n ) ∈ ( α , L ) ⇔ ( x k , α ( ( x k ) n ) ∈ ( α , I ) ⇔ ( x k ) n ∈ I ⇔ ( x k , ( ( x k ) n ) ∈ ( α , I ) ⇔ ( α , I ) = ( α , I ) … ( 1 ) .

Similarly, if for some n ∈ ℕ and x _t  ∈ Λ, ( x t ) n ∉ L , then we have ( x t ) n ∉ L ⇔ ( x t , Ø ) ∈ ( α , L ) ⇔ ( x t , Ø ) ∈ ( α , I ) ⇔ ( x t ) n ∉ I . ⇔ ( x t , Ø ) ∈ ( α , I ) ⇔ ( α , I ) = ( α , I ) … ( 2 ) .

From (1) and (2), we have ( α , I ) = ( α , L ) ⇔ ( α , I ) = ( α , I ) .

Definition 3.6. Let Λ be a ring, I = { 0 ¯ } be the zero ideal of Λ and ( α , I ) be a soft ideal over Λ. Then, the soft set ( α , Λ ) is called the ideal-soft nil radical of ( α , I ) , defined by = { α ( 0 ¯ ) , r n = 0 ¯ Ø , r n ≠ 0 ¯ , where α : Λ → ℘ (Λ) for all r ∈ Λ and some n ∈ ℕ .

Example 3.7. Let ( ℤ 8 , + , . ) be a ring and ( α , ℤ 8 ) be a soft ideal over ℤ 8 . For some n ∈ ℕ , 0 ¯ n = 2 ¯ n = 4 ¯ n = 6 ¯ n = 0 ¯ . Therefore, ( α , ℤ 8 ) = { ( 0 ¯ , α ( 0 ¯ ) ) , ( 1 ¯ , Ø ) , ( 2 ¯ , α ( 0 ¯ ) ) , ( 3 , Ø ) , ( 4 ¯ , α ( 0 ¯ ) ) , ( 5 ¯ , Ø ) , ( 6 ¯ , α ( 0 ¯ ) ) , ( 7 ¯ , Ø ) } is an ideal-soft nil radical of ( α , ℤ 8 ) .

Definition 3.8. Let ( α , A ) be a soft ring over Λ and ( γ , I ) be a soft ideal of ( α , A ) . Then the soft set ( γ , A ) is called a soft ideal-radical of ( γ , I ) defined by γ ( r ) = { ( r , γ ( r ) ) , ( r n , γ ( r n ) ) ∈ ( γ , I ) ( r , Ø ) , ( r n , γ ( r n ) ∉ ( γ , I ) , where γ : A → ℘ (Λ) for all r ∈ A and some n ∈ ℕ .

The soft ideal-radical ( γ , A ) can be denoted by ( γ , I ) .

Example 3.9. Let Λ = ℤ , A = Z 6 and I = { 0 ¯ , 3 ¯ } . Let us consider the function α: A → ℘ ( Z ) given by ( α , ℤ 6 ) = { ( 0 ¯ , ℤ ) , ( 1 ¯ , { 0 ¯ , 1 ¯ } ) , ( 2 ¯ , 2 ℤ ) , ( 3 ¯ , 3 ℤ ) , ( 4 ¯ , { 0 ¯ , 1 ¯ , 2 ¯ , 3 ¯ } ) , ( 5 ¯ , 5 ℤ ) } .

Let γ (x) equal to be { 0 ¯ } for all x ∈ I , where γ : I → ℘ ( ℤ 6 ) . The ( γ , I ) is soft ideal of ( α , ℤ 6 ) since { 0 ¯ } is ideal of ℤ and 3 ℤ . For some n ∈ ℕ , we find 0 ¯ n = 0 ¯ and 3 ¯ n = 3 ¯ ∈ I where γ : ℤ 6 → ℘ ( ℤ ) so ( 0 ¯ , γ ( 0 ¯ ) ) , ( 3 ¯ , γ ( 3 ¯ ) ) ∈ ( γ , ℤ 6 ) . For some n ∈ ℕ , 1 ¯ n , 2 ¯ n , 4 ¯ n and 5 ¯ n ∉ I so ( 1 ¯ , Ø ) , ( 2 ¯ , Ø ) , ( 4 ¯ , Ø ) and ( 5 ¯ , Ø ) ∈ ( γ , ℤ 6 ) . Hence, ( ( γ , ℤ 6 ) = { ( 0 ¯ , { 0 } ) , ( 1 ¯ , Ø ) , ( 2 ¯ , Ø ) , ( 3 ¯ , { 0 ¯ } ) , ( 4 ¯ , Ø ) , ( 5 ¯ , Ø ) } is obtained to be a soft ideal-radical of ( γ , I ) .

Definition 3.10. Let ( α , A ) be a soft ring over Λ, I = { 0 ¯ } be the zero ideal of Λ and ( γ , I ) be a soft ideal of ( α , A ) . The soft set ( γ , A ) is called a soft-nil radical of ( γ , I ) , defined by γ ( r ) = { ( r , γ ( 0 ¯ ) ) , r n = 0 ¯ ( r , Ø ) , r n ≠ 0 ¯ , where γ : A → ℘ ( Λ ) for all r ∈ Λ and some n ∈ ℕ .

Example 3.11. Let Λ = A = ( ℤ 8 , + , . ) be the ring and α be a function defined by α ( x ) = ℤ 8 for all x ∈ A . Then ( α , A ) is a soft ring over ℤ 8 . Let γ be a function defined by γ ( y ) = { 0 ¯ } for all y ∈ I = { 0 ¯ } . . Then ( γ , I ) is a a soft ideal of ( α , A ) . Hence, ( γ , ℤ 8 ) = { ( 0 ¯ , { 0 ¯ } ) , ( 1 ¯ , Ø ) , ( 2 ¯ , { 0 ¯ } ) , ( 3 ¯ , Ø ) , ( 4 ¯ , { 0 ¯ } ) , ( 5 ¯ , Ø ) , ( 6 ¯ , { 0 ¯ } ) , ( 7 ¯ , Ø ) } . is soft-nil radical of ( γ , I ) .

Theorem 3.12. Let ( α , A ) be a soft ring over a ring Λ, ( γ , I ) be a soft ideal of ( α , A ) and ( γ , I ) be soft ideal radical of ( γ , I ) . Then, ( γ , I ) ⊇ ( γ , I ) .

Proof. Suppose that ( α , A ) is soft ring over Λ and ( γ , I ) is soft ideal of ( α , A ) . It is obvious that ( γ , I ) is not empty.

For all r ∈ A , we can see that ( r , γ ( r ) ) ∈ ( γ , I ) and ( r , γ ( r ) ) ∈ ( γ , I ) , then r ∈ I . Since r ∈ I and rn ∈ I , we have ( r , γ ( r ) ) ∈ ( γ , I ) . Therefore ( γ , I ) ⊇ ( γ , I ) .

Theorem 3.13. Let ( α , A ) be a soft ring over a ring Λ and ( γ , I ) be a soft ideal of ( α , A ) . Then there exists a soft ideal ( γ , L ) of ( α , A ) such that ( γ , I ) = ( γ , L ) ⇔ ( γ , I ) = ( γ , I ) .

Proof. Suppose that ( γ , L ) is a soft ideal such that ( γ , I ) = ( γ , L ) . Let ( x k , γ ( x k ) ) ∈ ( γ , L ) . If ( x k ) n ∈ L for x k ∈ L and some n ∈ ℕ , then ( x k , γ ( x k ) ) ∈ ( γ , L ) ⇔ ( x k , γ ( x k ) ) ∈ ( γ , I ) ⇔ ( x k , γ ( x k ) ) ∈ ( γ , I ) ...   (3).

If ( x t ) n ∉ L for x t ∈ A and some n ∈ ℕ , then ( x t , Ø ) ∈ ( γ , L ) ⇔ ( x t , Ø ) ∈ ( γ , I ) ⇔ ( x t , Ø ) ∈ ( γ , I )  ...   (4).

From (3) and (4), we have ( γ , I ) = ( γ , L ) ⇔ ( γ , I ) = ( γ , I ) .

Now, we define radical of an ideal and introduce radical of a soft ideal in semigroup S .

Definition 3.14. Let I be nonempty ideal of semigroup S . I = { x : x n ∈ I , ∀ x ∈ S , ∃ n ∈ ℕ } is called radical of I .

Definition 3.15. Let A be a nonempty subset of semigroup S and ( φ , A ) be a soft ideal of S . Then, the ( φ , S ) is called a soft radical of ( φ , A ) , defined by φ ( s ) = { ∩ φ ( s n ) , s n ∈ A , ∀ s ∈ S , n ∈ N S , s n ∉ A , ∀ s ∈ S , ∀ n ∈ N where φ : S → ℘ ( S ) . The soft radical ( φ , S ) can be denoted by ( φ , A ) .

Example 3.16. Let S = ( Z 6 , . ) be a semigroup and A = { 1 ¯ , 4 ¯ , 5 ¯ } be the subset of S . Consider the function φ : A → ℘ ( ℤ 6 ) as φ ( 1 ¯ ) = { 0 , ¯ 3 ¯ } , φ ( 4 ¯ ) = { 0 , ¯ 2 , ¯ 4 ¯ } and φ ( 5 ¯ ) = { 0 ¯ } . ( φ , A ) is soft ideal of S . Then, ( φ , ℤ 6 ) = { ( 0 ¯ , ℤ 6 ) , ( 1 ¯ , { 0 , ¯ 3 ¯ } ) , ( 2 ¯ , { 0 , ¯ 2 , ¯ 4 ¯ } ) , ( 3 ¯ , ℤ 6 ) , ( 4 ¯ , { 0 , ¯ 2 , ¯ 4 ¯ } ) , ( 5 ¯ , { 0 ¯ } ) } is soft radical of ( φ , A ) .

Since, there exists some n ∈ ℕ such that 1 ¯ n , 2 ¯ n , 4 ¯ n and 5 ¯ n ∈ A , but 0 ¯ , 3 ¯ ∉ A .

Theorem 3.17. Let ( φ , A ) be a soft ideal over semigroup S . If ( φ , A ) is a soft radical of ( φ , A ) , then ( φ , A ) ⊇ ( φ , A ) and ( φ , A ) is a soft ideal over S .

Proof. Suppose A is nonempty subset of semigroup S and ( φ , A ) is a soft ideal of S . Assume that U ∈ ℘ ( S ) then ( x , U ) ∈ ( φ , A ) for ∀ x ∈ A . Since ( φ , A ) is a soft ideal, U is ideal of S .

If x n ∈ A for some n ∈ ℕ then, ∩ φ ( x n ) = φ ( x ) ∩ φ ( x n ) ⊆ φ ( x ) = U ( n ≠ 1 ) .

If x n ∉ A for all n ∈ ℕ then φ ( x n ) = S .

Therefore, we obtain that ( x , U ) ∈ ( φ , A ) and we can see ( φ , A ) ⊇ ( φ , A ) . It is clear that ( φ , A ) is a soft ideal over S .

Definition 3.18. Let S be a monoid and I = { 0 ¯ } be the zero ideal of S , I = { s : s n = 0 ¯ , ∀ s ∈ S , ∃ n ∈ ℕ } is called nil radical of I .

Definition 3.19. Let S be a monoid, I = { 0 ¯ } be the zero ideal of S and ( φ , I ) be a soft ideal. Then, the soft set ( φ , S ) is called the soft nil radical of ( φ , I ) , defined by φ ( s ) = { φ ( 0 ¯ ) , s n = 0 ¯ S , s n ≠ 0 ¯ , where φ : S → ℘ ( S ) for all s ∈ S and some n ∈ ℕ .

Example 3.20. Let S = ( Z 6 , + ) be monoid and I = { 0 ¯ } be the zero ideal of S . Consider the function φ : I → ℘ ( ℤ 6 ) defined by φ ( 0 ) = ℤ 6 . Then ( φ , I ) is soft ideal of S . Hence, ( φ , ℤ 6 ) = { ( 0 ¯ , ℤ 6 ) , ( 1 ¯ , ℤ 6 ) , ( 2 ¯ , ℤ 6 ) , ( 3 ¯ , ℤ 6 ) , ( 4 ¯ , ℤ 6 ) , ( 5 ¯ , ℤ 6 ) } is soft nil radical of ( φ , I ) .

For all s ∈ S , there exists some n ∈ ℕ such that s n = 0 ¯ , so ( s , ℤ 6 ) ∈ ( φ , ℤ 6 ) .

4 Discussion

The aim of this study was to construct a new algebraic structure on soft set theory which has paved the way for many studies. Throughout this paper, a radical for a soft ideal of a ring, a radical for a soft ideal of a soft ring and a radical for a soft ideal of a semigroup are characterized by the properties of these radicals. While the ideal-soft radical obtained from a ring provides radical theorems in general terms, the soft-ideal radical obtained from a soft ring remains within a limited framework. In addition, the soft radical structure was transferred to the ideal of a semigroup and nil radicals of all structures were obtained. Based on these results, some further works can be developed on the properties of the soft radicals for quasi ideals and bi-ideals in semigroups.