A type of strongly regular rings

1 Introduction

The study of fuzzy algebraic structure has been pursued in many directions such as groups, rings, modulus, vector spaces, topology and so on. In 1965, the American cyberneticist L.A. Zadeh [8] introduced the concept of fuzzy subsets and studied their properties on the parallel lines to set theory. In 1971, Rosenfeld [5] defined the fuzzy subgroup and gave some of its properties. Rosenfeld’s definition of a fuzzy group is a turning point for pure Mathematicians. In the definition of fuzzy subgroups, Rosenfeld has assumed that the subsets of a group G are fuzzy and the binary operation on G is nonfuzzy in the classical sense. Another approach is to assume that the set is nonfuzzy or classical and the binary operation is fuzzy in the fuzzy sense. Deviating a little away from this approach, Demirci [3, 4] introduced the concept of smooth group by the use of fuzzy binary operation and the concept of fuzzy equality, and Aktas [1] studied its application using Demirci’s concept.

So far fuzzy sets and fuzzy subgroup were defined on a given group G. Yuan and Lee [7] defined fuzzy binary operation on a set G. Aktas and Cagman [2] defined two fuzzy binary operation on a set R and defined fuzzy ring. In this paper our aim is to introduce the well known definition of (A : B) in a fuzzy ring R. In a ring (R, + , .) if A and B are subsets of R then (A : B) = {x ∈ R ∣ xB ⊆ A}. It has been shown that (A : B) is an ideal of the ring R if A and B are ideals of the ring R. We extend this notation as (A : B) = {x ∈ R ∣ ⋁ _b∈BH (x, b, y) > θ ⇒ y ∈ A} . We extend all the results obtained for (A : B) for classical ring to (A : B) in a fuzzy ring R.

2 Preliminaries

Let θ ∈ [0, 1). In this section the preliminarydefinition is summarized for understanding the concepts better; that is required in this paper. Most of the contents of this section are contained in [2, 7].

Definition 2.1. [7, Definition 2.2] Let G be a nonempty set and R be a fuzzy subset of G × G × G (R : G × G × G ⟶ [0, 1]). R is called a fuzzy binary operation on G if

∀ a, b ∈ G, ∃ c ∈ G such that R (a, b, c) > θ

Let R be a fuzzy binary operation on G, then we have a mapping R : F ( G ) × F ( G ) → F ( G ) ( A : B ) ↦ R ( A , B ) where F (G) = {A|A : R → [0, 1]is a mapping} andR ( A , B ) ( c ) = ⋁ a , b ∈ G ( A ( a ) ∧ B ( b ) ∧ R ( a , b , c ) )(1)

Let A = {a} and B = {b}, and let R (A, B) be denoted as a ∘ b, then ( a ∘ b ) ( c ) = R ( a , b , c ) , ∀ c ∈ G(2) ( ( a ∘ b ) ∘ c ) ( z ) = ⋁ d ∈ G ( R ( a , b , d ) ∧ R ( d , c , z ) )(3) ( a ∘ ( b ∘ c ) ) ( z ) = ⋁ d ∈ G ( R ( b , c , d ) ∧ R ( a , d , z ) )(4)

Using the notation in Equations (2)–(4), we get the following.

Definition 2.2. [7, Definitin 2.3] Let G be a nonempty set and R be a fuzzy binary operation on G. (G, R) is called a fuzzy abelian group if the following conditions are true:

* (G1) ∀ a, b, c, z₁, z₂ ∈ G, ((a ∘ b) ∘ c) (z₁) > θ and (a ∘ (b ∘ c)) (z₂) > θ implies z₁ = z₂;

* (G2) ∃e₀ ∈ G such that (e₀ ∘ a) (a) > θ and (a ∘ e₀) (a) > θ for any a ∈ G (e₀ is called an identity element of G);

* (G3) ∀ a ∈ G, ∃ b ∈ G such that (a ∘ b) (e₀) > θ and (b ∘ a) (e₀) > θ (b is called an inverse element of a and is denoted as a^-1).

* (G4) ∀ a, b ∈ G, (a ∘ b) (z₁) > θ and (b ∘ a) (z₂) > θ implies z₁ = z₂.

Proposition 2.1. [7, Proposition 2.1] Let (G, R) be a fuzzy group, then

the identity element of G is unique;

Definition 2.3. [2, Section 3] Let G and H be a fuzzy binary operations on R. Then we get a mapping as: G : F ( R ) × F ( R ) → F ( R ) ( A , B ) ↦ G ( A , B ) where F (R) = {A|A : R → [0, 1] is a mapping} and G ( A , B ) ( c ) = ⋁ a , b ∈ R ( A ( a ) ∧ B ( b ) ∧ R ( a , b , c ) )

Let A = {a} and B = {b}, and let G (A, B) and H (A, B) be denoted as a ∘ b and a ∗ b respectively. Then ( a ∘ b ) ( c ) = G ( a , b , c ) , ∀ c ∈ R(5) ( a ∗ b ) ( c ) = H ( a , b , c ) , ∀ c ∈ R(6) ( ( a ∘ b ) ∘ c ) ( z ) = ⋁ d ∈ G ( G ( a , b , d ) ∧ G ( d , c , z ) )(7) ( a ∘ ( b ∘ c ) ) ( z ) = ⋁ d ∈ G ( G ( b , c , d ) ∧ G ( a , d , z ) )(8) ( a ∗ ( b ∘ c ) ) ( z ) = ⋁ d ∈ G ( G ( b , c , d ) ∧ H ( a , d , z ) )(9) ( ( a ∗ b ) ∘ ( a ∗ c ) ) ( z ) = ⋁ d ∈ G ( H ( a , b , d ) ∧ H ( a , c , e ) ∧ G ( d , e , z ) )(10)

Using the notation of equations (5)–(10), we candefine a kind of fuzzy ring as:

Definition 2.4. [2, Definition 9] Let R be a nonempty set, and G and H be two fuzzy binary operation on R. Then (R, G, H) is called a fuzzy ring if the following conditions hold.

(R, G) is an abelian fuzzy group.

The identity element e₀ is called the zero element of the fuzzy ring.

A fuzzy ring has the following properties:

Theorem 2.1. If e₀ is the zero element of (R, G) in the fuzzy ring (R, G, H) then for any a ∈ R, H (a, e₀, x) > θ ⇒ x = e₀

Proof. Let a, z ∈ R. Now ( a ∗ ( e 0 ∘ e 0 ) ) ( z ) = ⋁ d ∈ R ( G ( e 0 , e 0 , d ) ∧ H ( a , d , z ) ) ( ( a ∗ e 0 ) ∘ ( a ∗ e 0 ) ) ( z ) = ⋁ d , f ∈ R ( H ( a , e 0 , d ) ∧ H ( a , e 0 , f ) ∧ G ( d , f , z ) )

Let x ∈ R such that H (a, e₀, x) > θ.

Now [a ∗ (e₀ ∘ e₀)] (x) > θ since H (a, e₀, x) > θ and G (e₀, e₀, e₀) > θ

Hence ((a ∗ e₀) ∘ (a ∗ e₀)) (x) > θ

Thus H (a, e₀, x) ∧ H (a, e₀, x) ∧ G (x, x, x) > θ.

Hence G (x, x, x) > θ

Therefore x = e₀ by proposition 2.1

Thus H (a, e₀, e₀) > θ

Theorem 2.2. If (R, G, H) is a fuzzy ring with zero element e₀, then for any a, b ∈ R:

Let b^-1 be the inverse of b in (R, G). Then H (a, b, x) > θ ⇔ H (a, b^-1, x^-1) > θ.

Definition 2.5.[2, Definition 18] A nonempty subset I of a fuzzy ring R is called a fuzzy left (resp. right) ideal of R if the following conditions are satisfied.

∀ x, y ∈ I, (x ∘ y) (z) > θ ⇒ z ∈ I for z ∈ R

Definition 3.1. Let A and B be any subset of R then ( A : B ) = { x ∈ R ∣ ⋁ b ∈ B H ( x , b , y ) > θ ⇒ y ∈ A }

Theorem 3.1. Let (R, G, H) be a fuzzy ring and A be any fuzzy ideal of R and B be any subset of R then (A: B) is a fuzzy left ideal of (R, G, H).

Proof. 1. Let x₁, x₂ ∈ (A : B). Suppose (x₁ ∘ x₂) (z) > θ, let us show that z ∈ (A : B). Since (x₁ ∘ x₂) (z) > θ, G ( x 1 , x 2 , z ) > θ(11)

Now, suppose ⋁ b ∈ B H ( z , b , y ) > θ

Hence there exist b₁ ∈ B such thatH ( z , b 1 , y ) > θ(12)

Now consider ( ( x 1 ∘ x 2 ) ∗ b 1 ) ( y ) = ⋁ d ∈ R ( G ( x 1 , x 2 , d ) ∧ H ( d , b 1 , y ) ) ≥ G ( x 1 , x 2 , z ) ∧ H ( z , b 1 , y ) > θ ( from ( 11 ) and ( 12 ) )

Therefore ((x₁ ∘ x₂) ∗ b₁) (y) > θ

Thus ((x₁ ∗ b₁) ∘ (x₂ ∗ b₁)) (y) > θ

Hence ⋁ d , e ∈ R ( H ( x 1 , b 1 , d ) ∧ H ( x 2 , b 1 , e ) ∧ G ( d , e , y ) ) > θ

By definition there exists unique d₁ ∈ R and unique e₁ ∈ R such that H (x₁, b₁, d₁) > θ, H (x₂, b₁, e₁) > θ . Hence G (d₁, e₁, y) > θ

Since x₁ ∈ (A : B) and H (x₁, b₁, d₁) > θ ⇒ d₁ ∈ A

Since x₂ ∈ (A : B) and H (x₂, b₁, e₁) > θ ⇒ e₁ ∈ A

Since d₁, e₁ ∈ A and A is fuzzy ideal of R and G (d₁, e₁, y) > θ ⇒ y ∈ A

Thus ⋁ b ∈ B H ( z , b , y ) > θ ⇒ y ∈ A

Therefore z ∈ (A : B).

2. Let x ∈ (A : B). Let us show that x^-1 ∈ (A : B).Suppose ⋁ b ∈ B H ( x - 1 , b , y ) > θ .

Hence there exists b₁ ∈ B such that H (x^-1, b₁, y) > θ . By Theorem 2.2, H (x, b₁, y^-1) > θ ⇒ y^-1 ∈ A . Since A is an ideal, y ∈ A. Therefore x^-1 ∈ (A : B).

3. Let x ∈ (A : B) and let r ∈ R. Suppose (r ∗ x) (z) > θ H ( r , x , z ) > θ(13)

Let us show that z ∈ (A : B).

Suppose ⋁ b ∈ B H ( z , b , y ) > θ

Hence there exists b₁ ∈ B such thatH ( z , b 1 , y ) > θ(14)

Now [ ( r ∗ x ) ∗ b 1 ] ( y ) = ⋁ d ∈ R ( H ( r , x , d ) ∧ H ( d , b 1 , y ) ) ≥ H ( r , x , z ) ∧ H ( z , b 1 , y ) > θ ( from ( 13 ) and ( 14 ) )

Hence (r ∗ (x ∗ b₁)) (y) > θ

Therefore ⋁ d ∈ R ( H ( x , b 1 , d ) ∧ H ( r , d , y ) ) > θ

By definition there exists unique d₁ ∈ R such that H (x, b₁, d₁) > θ and hence H (r, d₁, y) > θ .

Since x ∈ (A : B) and H (x, b₁, d₁) > θ implies d₁ ∈ A .

Since H (r, d₁, y) > θ implies y ∈ A .

Therefore z ∈ (A : B) .

(r ∗ x) (z) > θ implies z ∈ (A : B) .

Therefore (A : B) is a fuzzy left ideal.

Theorem 3.2. If A and B are fuzzy ideals of (R, G, H) then (A : B) is a fuzzy ideal of (R, G, H).

Proof. By Theorem 3.1, (A : B) is a fuzzy left ideal.

Let x ∈ (A : B) and r ∈ R .

Suppose (x ∗ r) (z) > θ .

HenceH ( x , r , z ) > θ(15)Let us show that z ∈ (A : B).

Suppose ⋁ b ∈ B H ( z , b , y ) > θ Hence there exists b₁ ∈ B such thatH ( z , b 1 , y ) > θ(16)Now [ ( x ∗ r ) ∗ b 1 ] ( y ) = ⋁ d ∈ R ( H ( x , r , d ) ∧ H ( d , b 1 , y ) ) ≥ H ( x , r , z ) ∧ H ( z , b 1 , y ) > θ ( from ( 15 ) and ( 16 ) ) Hence (x ∗ (r ∗ b₁)) (y) > θ

Therefore ⋁ d ∈ R ( H ( r , b 1 , d ) ∧ H ( x , d , y ) ) > θ By definition there exists d₁ ∈ R such that H (r, b₁, d₁) > θ and hence H (x, d₁, y) > θ . Since B is an ideal and H (r, b₁, d₁) > θ implies d₁ ∈ B .

Since d₁ ∈ B and H (x, d₁, y) > θ implies y ∈ A.

Thus z ∈ (A : B) .

(x ∗ r) (z) > θ implies z ∈ (A : B)

Therefore (A : B) is a fuzzy ideal

Definition 3.2. For any a ∈ R, the left annihilator of a in R is l (a) = {x ∈ R ∣ H (x, a, y) > θ implies y = e₀}

Theorem 3.3. For any a ∈ R, the left annihilator of a is a left ideal.

Proof. In Theorem 3.1, by taking A = {e₀} and B = {a} we get (A : B) = l (a) and hence l (a) is a left ideal.

We have proved that the left annihilator of “a” is the left ideal by using the fact that (A : B) is a left ideal. In general, left annihilator of " a " is not an ideal, but when the ring R is strongly regular then it is proved that the left annihilator of " a " is an ideal.

Definition 4.1. A fuzzy ring (R, G, H) is said to be regular if for each a in (R, G, H) there exists x ∈ (R, G, H) such that ((a ∗ x) ∗ a) (a) > θ.

Theorem 4.1. Let (R, G, H) be a regular ring and I be an ideal of (R, G, H). Then for any a ∈ I, we can find x, y ∈ I such that (x ∗ y) (a) > θ.

Proof. Let I be an ideal of R and let a ∈ I. Since R is regular there exists x ∈ R such that ((a ∗ x) ∗ a) (a) > θ

Hence, ⋁ d ∈ R ( H ( a , x , d ) ∧ H ( d , a , a ) ) > θ

By definition of H there exists unique d₁ ∈ R such that H (a, x, d₁) > θ and since a ∈ I implies d₁ ∈ I .

Hence H (d₁, a, a) > θ where d₁ ∈ I .

Therefore (d₁ ∗ a) (a) > θ where d₁, a ∈ I .

Definition 4.2. A fuzzy ring (R, G, H) is said to be left strongly regular if for each a, b ∈ (R, G, H) there exists x ∈ (R, G, H) such that (a ∗ b) (z) > θ ⇔ (x ∗ (a ∗ b) ∗ (a ∗ b)) (z) > θ.

Theorem 4.2. Let (R, G, H) be a fuzzy ring and e_∗ be the identity element of (R, H) and for any a, b in R, ((a ∗ e_∗) ∗ b) (z) > θ ⇒ (a ∗ b) (z) > θ.

Proof. Assume ((a ∗ e_∗) ∗ b) (z) > θ. Hence ⋁ f ∈ R ( H ( a , e ∗ , f ) ∧ H ( f , b , z ) ) > θ We must have f = a and H (a, b, z) > θ .

Therefore (a ∗ b) (z) > θ.

Theorem 4.3. If R is left strongly regular then given a ∈ (R, G, H) there exists x ∈ (R, G, H) such that (x ∗ (a ∗ a)) (a) > θ.

Proof. Assume R is left strongly regular.

Let a ∈ (R, G, H) and e_∗ be the identity element of (R, H). Now take b as e_∗ . Then there exists x ∈ R such that (a * e_∗) (z) > θ ⇔ (x ∗ (a ∗ e_∗) ∗ (a ∗ e_∗)) (z) > θ .

Since (a ∗ e_∗) (a) > θ we have (x ∗ (a ∗ e_∗) ∗ (a ∗ e_∗)) (a) > θ By Theorem 4.2. we have (x ∗ a ∗ a) (a) > θ.

Theorem 4.4. Let I be an ideal of a strongly regular ring (R, G, H). For any a ∈ R,if H (a, a, y) > θ implies y ∈ I then a ∈ I.

Proof. Let I be an ideal of R and let a ∈ R. Since R is left strongly regular by Theorem 4.3, there exists x ∈ R such that (x ∗ (a ∗ a)) (a) > θ . Hence⋁ d ∈ R ( H ( a , a , d ) ∧ H ( x , d , a ) ) > θ(17)

Hence there exists unique d₁ ∈ R such that H (a, a, d₁) > θ and H (x, d₁, a) > θ.

By assumption H (a, a, d₁) > θ implies d₁ ∈ I. Thus H (x, d₁, a) > θ with d₁ ∈ I implies a ∈ I .

Theorem 4.5. Let I be an ideal of a fuzzy ring (R, G, H) and for any a, b, c ∈ R,if (a ∗ b) (y) > θ ⇒ y ∈ I then ((a ∗ b) ∗ c) (z) > θ ⇒ z ∈ I.

Proof. Suppose (a ∗ b) (y) > θ ⇒ y ∈ I .

Thus H (a, b, y) > θ ⇒ y ∈ I .

Assume ((a ∗ b) ∗ c) (z) > θ .

Now⋁ f ∈ R ( H ( a , b , f ) ∧ H ( f , c , z ) ) > θ(18)By definition of H there exists unique d ∈ R such that H (a, b, d) > θ and hence we have H (d, c, z) > θ .

Since H (a, b, d) > θ implies d ∈ I .

Since d ∈ I and H (d, c, z) > θ implies z ∈ I .

Therefore ((a ∗ b) ∗ c) (z) > θ implies z ∈ I .

Theorem 4.6. Let I be an ideal of a fuzzy ring (R, G, H) and for any a, b, c ∈ R, if (a ∗ b) (y) > θ ⇒ y ∈ I then (c ∗ (a ∗ b)) (z) > θ ⇒ z ∈ I.

Proof is similar to Theorem 4.5.

Theorem 4.7. Let I be an ideal of a strongly regular ring (R, G, H) and let x ∈ R for any a, b ∈ I such that H (a, b, x) > θ ⇒ x ∈ I then H (b, a, y) > θ ⇒ y ∈ I.

Proof. Since R is a left strongly regular, there exists x ∈ R such that (a ∗ b) (z) > θ ⇔ (x ∗ (a ∗ b) ∗ (a ∗ b)) (z) > θ . Let H (b, a, y) > θ. Hence (b ∗ a) (y) > θ using Theorem 4.5 and 4.6 we have ((b ∗ a) ∗ b) (y) > θ and hence (x ∗ (a ∗ b) ∗ (a ∗ b)) (y) > θ. Hence (a ∗ b) (y) > θ . Thus y ∈ I .

Theorem 4.8. If e₀ is the zero element of (R, G) in the fuzzy ring (R, G, H) then for any a, b, c ∈ R, (a ∗ b) (e₀) > θ ⇒ (c ∗ (a ∗ b)) (e₀) > θ and ((a ∗ b) ∗ c) (e₀) > θ .

Proof. Let a, b, c be in R such thatH ( a , b , e 0 ) > θ(19)Now ( c ∗ ( a ∗ b ) ) ( e 0 ) = ⋁ d ∈ R ( H ( a , b , d ) ∧ H ( c , d , e 0 ) ) ≥ H ( a , b , e 0 ) ∧ H ( c , e 0 , e 0 ) > θ ( by ( 19 ) and Theorem 2.1 ) Similarly ((a ∗ b) ∗ c)) (e₀) > θ .

Definition 4.3. An element a of a ring R is called a nilpotent element if there exists some positive integer n such that (a ∗ a ∗ a ∗ . . . ∗ a) (e₀) > θ implies a = e₀.

Theorem 4.9. If (R, G, H) is a left strongly regular then (R, G, H) is without non-zero nilpotent elements.

Proof. Suppose (a ∗ a) (e₀) > θ . Since R is a left strongly regular there eists x ∈ R such that( x ∗ ( a ∗ a ) ) ( a ) > θ(20)

By Theorem 4.6, since (a ∗ a) (e₀) > θ implies( x ∗ ( a ∗ a ) ) ( e 0 ) > θ(21)

From (20) and (21), a = e₀

Theorem 4.10. Suppose (R, G, H) is a left strongly regular. If (a ∗ b) (e₀) > θ for some a, b ∈ R then (b ∗ a) (e₀) > θ .

Proof. GivenH ( a , b , e 0 ) > θ(22)By Theorem 4.6 (b ∗ a ∗ b ∗ a) (e₀) > θ . Since (R, G, H) is a left strongly regular there exists   x ∈ R such that (x ∗ (b ∗ a) ∗ (b ∗ a)) (e₀) > θ. Therefore (b ∗ a) (e₀) > θ

Definition 4.4. A fuzzy ring R is said to fulfill the insertion of factors property (IFP) provided that for all a, b ∈ R, (a ∗ b) (e₀) > θ ⇒ (a ∗ r ∗ b) (e₀) > θ for all r ∈ R .

Theorem 4.11. Let (R, G, H) be a left strongly regular fuzzy ring. Then for any a, b ∈ R, (a ∗ b) (e₀) > θ ⇒ (a ∗ r ∗ b) (e₀) > θ for all r ∈ R.

Proof. Given (a ∗ b) (e₀) > θ . By Theorem 4.7 (b ∗ a) (e₀) > θ . Let r ∈ R using Theorem 4.6 we have (r ∗ b ∗ a) (e₀) > θ and hence (a ∗ r ∗ b ∗ a ∗ r ∗ b) (e₀) > θ . Then there exists x ∈ R such that (x ∗ a ∗ r ∗ b ∗ a ∗ r ∗ b) (e₀) > θ . Since R is left strongly regular (a ∗ r ∗ b) (e₀) > θ . Therefore R has insertion factors property.

Theorem 4.12. If (R, G, H) is left strongly regular, then for any a ∈ R, left annihilator of “a” is an ideal.

Proof. By Theorem 3.3, l (a) is a left ideal. Let x ∈ l (a) and r ∈ R,H ( x , a , e 0 ) > θ(23)Let us show that (x ∗ r ∗ a) (e₀) > θ . Since (x ∗ a) (e₀) > θ, By Theorem 4.4, R has insertion factors property, (x ∗ r ∗ a) (e₀) > θ.

In Section 3, we have proved that if A and B are fuzzy ideals of (R, G, H) then (A : B) is an ideal of (R, G, H). In Section 4, we have shown that if I is an ideal of a strongly regular ring (R, G, H) and let x ∈ I for any a, b ∈ I such that H (a, b, x) > θ ⇒ x ∈ I then H (b, a, y) > θ ⇒ y ∈ I. We also prove that if (R, G, H) is strongly regular, then for any a ∈ R, left annihilator of a is an ideal.

For further investigation, characterization of prime ideals, quasi-ideal and bi-ideal may be investigated. One can extend the results obtained fuzzy ring to fuzzy ternary ring.