Characterizations of non-associative rings by the properties of their fuzzy ideals

In this paper, we extend the characterizations of Kuroki [Regular fuzzy duo rings. Inform Sci. 1996;96:119–139], by initiating the concept of fuzzy left (resp. right, interior, quasi-, bi-, generalized bi-) ideals in a class of non-associative and non-commutative rings (LA-ring). We characterize regular (intra-regular, both regular and intra-regular) LA-rings in terms of such ideals.


Introduction
In ternary operations, the abelian law is given by abc = cba. Kazim et al. [9] have generalized this notion by introducing the parenthesis on the left side of this equation abc = cba to get a new pseudo associative law, that is (ab)c = (cb)a. This law (ab)c = (cb)a is called the left invertive law. A groupoid S is left almost semigroup (abbreviated as LA-semigroup ), if it satisfies the left invertive law. An LA-semigroup is a midway structure between a abelian semigroup and a groupoid. Ideals in LA-semigroups have been investigated by Protic et al. [16].
The notion of LA-semigroup is extended to the left almost group (abbreviated as LA-group) by Kamran [5]. An LA-semigroup S is left almost group, if there exists a left identity e ∈ S such that ea = a for all a ∈ S and for every a ∈ S there exists b ∈ S such that ba = e.
Shah et al. [19] discussed the left almost ring (abbreviated as LA-ring) of finitely nonzero functions, which is a generalization of commutative semigroup ring. By a left almost 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. Although the structure is non-associative and noncommutative, nevertheless, it possesses many interesting properties which we usually find in associative and commutative algebraic structures.
A non-empty subset A of an LA-ring R is an LAsubring of R if a−b and ab ∈ A for all a, b ∈ A. A is a left (resp. right) ideal of R if (A, +) is an LA-group and RA ⊆ A (resp. AR ⊆ A). A is an ideal of R if it is both a left ideal and a right ideal of R.
A non-empty subset A of an LA-ring R is interior ideal of R, if (A, +) is an LA-group and (RA)R ⊆ A. A nonempty subset A of an LA-ring R is quasi-ideal of R, if (A, +) is an LA-group and AR ∩ RA ⊆ A. An LA-subring We define the concept of fuzzy left (resp. right, interior, quasi-, bi-, generalized bi-) ideals of an LA-ring R. We will establish a study by discussing the different properties of such ideals. We will also characterize regular (resp. intra-regular, both regular and intra-regular) LAings by the properties of fuzzy left (right, quasi-, bi-, generalized bi-) ideals. provided a useful mathematical tool for describing the behaviour of systems that are too complex to admit precise mathematical analysis by classical methods and tools. Extensive applications of fuzzy set theory have been found in various fields such as artificial intelligence, computer science, management science, expert systems, finite state machines, Languages, robotics, coding theory and others.
It soon invoked a natural question concerning a possible connection between fuzzy sets and algebraic systems like (set, group, semigroup, ring, near-ring, semiring, measure) theory, groupoids, real analysis, topology, differential equations and so forth. Rosenfeld [18], was the first, who introduced the concept of fuzzy set in a group. The study of fuzzy set in semigroup was established by Kuroki [10]. He studied fuzzy ideals and fuzzy (interior, quasi-, bi-, generalized bi-, semiprime) ideals of semigroups.
Liu [12] introduced the concept of fuzzy subrings and fuzzy ideals of a ring. Many authors have explored the theory of fuzzy rings [2,3,11,13,14,22]. Gupta et al. [3] gave the idea of intrinsic product of fuzzy subsets of a ring. Kuroki [11] characterized regular (intra-regular, both regular and intra-regular) rings in terms of fuzzy left (right, quasi, bi-) ideals.
Kausar et al. [20] initiated the concept of intuitionistic fuzzy normal subrings over a non-associative ring and also characterized the non-associative rings by their intuitionistic fuzzy bi-ideals in [6]. Recently Kausar [8] explored the notion of direct product of finite intuitionistic anti-fuzzy normal subrings over non-associative rings.
A fuzzy subset μ of an LA-ring R is a function from R into the closed unit interval [0, 1], that is μ : R → [0, 1], the complement of μ is denoted by μ , is also a fuzzy subset of R defined by μ (x) = 1 − μ(x) for all x ∈ R. F(R) denotes the collection of all fuzzy subsets of R.
A fuzzy subset μ of an LA-ring R is a fuzzy LA- for all x, y ∈ R. μ is called a fuzzy ideal of R if it is both a fuzzy left ideal and a fuzzy right ideal of R.
Let A be a non-empty subset of an LA-ring R. Then the characteristic function of A is denoted by χ A and defined by We note that an LA-ring R can be considered a fuzzy subset of itself and we write R = C R , i.e. R(x) = 1 for all x ∈ R. Let μ and γ be two fuzzy subsets of an LA- The product of μ and γ is denoted by μ • γ and defined by A fuzzy subset μ of an LA-ring R is a fuzzy interior ideal <script type= " math/tex " id= " Now we give the imperative properties of such ideals of an LA-ring R, which will play a vital rule in the later sections.
Lemma 2.1: Let R be an LA-ring. Then the following properties hold.
Proof: Let μ, γ and β be fuzzy subsets of an LA-ring R.
Similarly, we can prove (2). Proposition 2.1: Let R be an LA-ring with left identity e. Then the following assertions hold.
Then there are two cases.
(ii) If x = uv for some u, v ∈ R, then (3) Let a ∈ R and assume that a ∈ A ∪ B. Then there are three cases.
(i) If a ∈ A and a ∈ B, then (ii) If a ∈ A and a / ∈ B, then (iii) If a / ∈ A and a ∈ B, then If a / ∈ A ∪ B, this implies that a / ∈ A and a / ∈ B. Then obvious χ A ∪ χ B = χ A∪B . Hence in all cases χ A ∪ χ B = χ A∪B .
(4) Let a ∈ R and suppose that a ∈ A ∩ B, this means that a ∈ A and a ∈ B. Now ( Since a−b and ab ∈ A, A being an LA-subring of R, this implies that χ A (a − b) = 1 and χ A (ab) = 1.
Conversely, suppose that χ A is a fuzzy LA-subring of R and let a, b ∈ A. (1) μ is a fuzzy LA-subring of R if and only if μ • μ ⊆ μ and μ − μ ⊆ μ.
for all x, y ∈ R.
Proof: (1) Suppose that μ is a fuzzy LA-subring of R and Conversely, assume that μ • μ ⊆ μ and μ − μ ⊆ μ. Let x and y ∈ R such that a = xy. Now Hence μ is a fuzzy LA-subring of R.
(2) Suppose that μ is a fuzzy left ideal of R and x ∈ R.
Therefore μ is a fuzzy left ideal of R. Similarly, we can prove (3).
Proof: Let μ and γ be two fuzzy LA-subrings of R. We have to show that μ ∩ γ is also a fuzzy LA-subring of R. Now Similarly, for ideals.

Lemma 2.3:
If μ and γ are two fuzzy LA-subrings of an LA-ring R, then μ • γ is also a fuzzy LA-subring of R.
Proof: Let μ and γ be two fuzzy LA-subrings of R.

Remark 2.2:
If μ is a fuzzy LA-subring of an LA-ring R, then μ • μ is also a fuzzy LA-subring of R.

Lemma 2.4:
Let R be an LA-ring with left identity e. Then RR = R and eR = R = Re.

Lemma 2.5: Let R be an LA-ring with left identity e. Then every fuzzy right ideal of R is a fuzzy ideal of R.
Proof: Let μ be a fuzzy right ideal of R and x, y ∈ R. Now Lemma 2.6: If μ and γ are two fuzzy left (resp. right) ideals of an LA-ring R with left identity e, then μ • γ is also a fuzzy left (resp. right) ideal of R.
Proof: Let μ and γ be two fuzzy left ideals of R. We have to show that μ • γ is also a fuzzy left ideal of R.

Remark 2.3:
If μ is a fuzzy left (resp. right) ideal of an LA-ring R with left identity e, then μ • μ is a fuzzy ideal of R. Lemma 2.7: If μ and γ are two fuzzy ideals of an LA-ring R, then μ • γ ⊆ μ ∩ γ .
Proof: Let μ and γ be two fuzzy ideals of R and

Remark 2.4:
If μ is a fuzzy ideal of an LA-ring R, then μ • μ ⊆ μ.

(1) A is an interior ideal of R if and only if χ A is a fuzzy interior ideal of R. (2) A is a quasi-ideal of R if and only if χ A is a fuzzy quasiideal of R. (3) A is a bi-ideal of R if and only if χ A is a fuzzy biideal of R. (4) A is a generalized bi-ideal of R if and only if χ A is a fuzzy
generalized bi-ideal of R.
Conversely, suppose that χ A is a fuzzy interior ideal of R, this implies that χ A is a fuzzy additive LAsubgroup of R. Then A is an additive LA-subgroup of R by Remark 2.1. Let x, y ∈ R and a ∈ A, so χ A (a) = 1. Since (2) Let A be a quasi-ideal of R, this implies that A is an additive LA-subgroup of R. Then χ A is a fuzzy additive LA-subgroup of R. Now Conversely, assume that χ A is a fuzzy quasi-ideal of R, this means that χ A is a fuzzy additive LA-subgroup of R. Then A is an additive LA-subgroup of R. Let x be an element of AR ∩ RA. Now Similarly, we can prove (4).

Theorem 2.5: Let μ be a fuzzy subset of an LA-ring R. Then μ is a fuzzy interior ideal of R if and only if
Therefore μ is a fuzzy interior ideal of R.

Theorem 2.6: Let μ be a fuzzy LA-subring of an LA-ring R. Then μ is a fuzzy bi-ideal of R if and only if
Proof: Same as Theorem 2.5.

Theorem 2.7: Let μ be a fuzzy subset of an LA-ring R. Then μ is a fuzzy generalized bi-ideal of R if and only if
Proof: Same as Theorem 2.5. Lemma 2.9: If μ and γ are two fuzzy bi-(resp. generalized bi-, quasi-, interior) ideals of an LA-ring R, then μ ∩ γ is also a fuzzy bi-(resp. generalized bi-, quasi-, interior) ideal of R.
Proof: Let μ and γ be two fuzzy bi-ideals of R. We have to show that μ ∩ γ is also a fuzzy bi-ideal of R. Now Hence μ ∩ γ is a fuzzy bi-ideal of R. Lemma 2.10: If μ and γ are two fuzzy bi-(resp. generalized bi-, interior) ideals of an LA-ring R with left identity e, then μ • γ is also a fuzzy bi-(resp. generalized bi-, interior) ideal of R.
Proof: Let μ and γ be two fuzzy bi-ideals of R. We have to show that μ • γ is also a fuzzy bi-ideal of R. Since μ and γ are fuzzy LA-subrings of R, then μ • γ is also a fuzzy LA-subring of R by the Lemma 2.3. Now Therefore μ • γ is a fuzzy bi-ideal of R. Proof: Let μ be a fuzzy interior ideal of R and x, y ∈ R. Now μ(xy) = μ((ex)y) ≥ μ(x), thus μ is a fuzzy right ideal of R. Hence μ is a fuzzy ideal of R by Lemma 2.5. Converse is true by Lemma 2.11.

Lemma 2.12: Every fuzzy left (resp. right, two-sided) ideal of an LA-ring R is a fuzzy bi-ideal of R. The converse is not true in general.
Proof: Suppose that μ is a fuzzy right ideal of R and x, y, z ∈ R. Now μ((xy)z) = μ(xy) ≥ μ(x) and μ((xy)z) = μ((zy)x) ≥ μ(zy) ≥ μ(z), this implies that μ((xy)z) ≥ μ(x) ∧ μ(z). Hence μ is a fuzzy bi-ideal of R. Lemma 2.13: Every fuzzy bi-ideal of an LA-ring R is a fuzzy generalized bi-ideal of R. The converse is not true in general.

Proof: Obvious.
Lemma 2.14: Every fuzzy left (resp. right, two-sided) ideal of an LA-ring R is a fuzzy quasi-ideal of R. The converse is not true in general.

Proposition 2.3: Every fuzzy quasi-ideal of an LA-ring R is a fuzzy LA-subring of R.
Proof: Let μ be a fuzzy quasi-ideal of R.
So μ is a fuzzy LA-subring of R.

Proposition 2.4:
Let μ be a fuzzy right ideal and γ be a fuzzy left ideal of an LA-ring R, respectively. Then μ ∩ γ is a fuzzy quasi-ideal of R. Proof: Let μ be a fuzzy quasi-ideal of R. Since μ • μ ⊆ μ by Proposition 2.3. Now Hence μ is a fuzzy bi-ideal of R. Proposition 2.5: If μ and γ are two fuzzy quasi-ideals of an LA-ring R with left identity e, such that (xe)R = xR for all x ∈ R. Then μ • γ is a fuzzy bi-ideal of R.
Proof: Let μ and γ be two fuzzy quasi-ideals of R, this implies that μ and γ be two fuzzy bi-ideals of R, by Lemma 2.15. Then μ • γ is also a fuzzy bi-ideal of R by Lemma 2.10.

Regular left almost rings
An LA-ring R is regular if for every x ∈ R, there exists an element a ∈ R such that x = (xa)x. In this section, we characterize regular LA-rings by the properties of fuzzy ( left, right, quasi-, bi-, generalized bi-) ideals.

Lemma 3.1: Every fuzzy right ideal of a regular LA-ring R is a fuzzy ideal of R.
Proof: Suppose that μ is a fuzzy right ideal of R. Let x, y ∈ R, this implies that there exists an element a ∈ R, such that x = (xa)x. Thus μ(xy) = μ(((xa)x)y) = μ((yx)(xa)) ≥ μ(yx) ≥ μ(y). Hence μ is a fuzzy ideal of R.

Lemma 3.2: Every fuzzy ideal of a regular LA-ring R is a fuzzy idempotent.
Proof: Assume that μ is a fuzzy ideal of R and μ • μ ⊆ μ. We have to show that μ ⊆ μ • μ. Let x ∈ R, this means that there exists an element a ∈ R such that x = (xa)x. Thus Therefore μ = μ • μ.

Remark 3.1: Every fuzzy right ideal of a regular LA-ring
R is a fuzzy idempotent.

Lemma 3.3: Let μ be a fuzzy subset of a regular LA-ring R. Then μ is a fuzzy ideal of R if and only if μ is a fuzzy interior ideal of R.
Proof: Suppose that μ is a fuzzy interior ideal of R. Let x, y ∈ R, then there exists an element a ∈ R, such that x = (xa)x. Thus μ(xy) = μ(((xa)x)y) = μ((yx)(xa)) ≥ μ(x), i.e. μ is a fuzzy right ideal of R. So μ is a fuzzy ideal of R by Lemma 3.1. Converse is true by Lemma 2.11.

Remark 3.2:
The concept of fuzzy (interior, two-sided) ideals coincides in regular LA-rings. Proof: Assume that μ is a fuzzy right ideal of R. Then (μ • R) ∩ (R • μ) ⊆ μ, because every fuzzy right ideal of R is a fuzzy quasi-ideal of R by Lemma 2.14. Let x ∈ R, this implies that there exists an element a ∈ R, such that Proof: Since μ • γ ⊆ μ ∩ γ for every fuzzy right ideal μ and every fuzzy left ideal γ of R by Lemma 2.8. Let x ∈ R, this means that there exists an element a ∈ R such that Hence μ • γ = μ ∩ γ .

Proposition 3.2 ([6, Proposition 5]): Let R be an LAring with left identity e. Then aR ∪ Ra is the smallest right ideal of R containing a.
(1) implies (3). Assume that (3) holds. Let μ be a fuzzy right ideal and γ be a fuzzy left ideal of R. This means that μ and γ be fuzzy quasi-ideals of R by Lemma 2.14, so μ ∩ γ be also a fuzzy quasi-ideal of R. Then by our assumption, (2) is true and a ∈ R. Then Ra is a left ideal of R containing a by Lemma 3.5 and aR ∪ Ra is a right ideal of R containing a by Proposition 3.2. So χ Ra is a fuzzy left ideal and χ aR∪Ra is a fuzzy right ideal of R, by Theorem 2.2. Then by our supposition So a is a regular, i.e. R is a regular. Hence (2) ⇒ (1).

Theorem 3.2:
Let R be an LA-ring with left identity e, such that (xe)R = xR for all x ∈ R. Then the following conditions are equivalent.
(1) R is a regular. (1) R is a regular.
• β for every fuzzy bi-ideal β and for every fuzzy ideal γ of R.

for every fuzzy generalized biideal δ and for every fuzzy ideal γ of R.
Proof: Suppose that (1) holds. Let δ be a fuzzy generalized bi-ideal and γ be a fuzzy ideal of R.
Thus (1) R is a regular.

for every fuzzy generalized bi-ideal δ, for every fuzzy right ideal γ and for every fuzzy left ideal λof R.
Proof: Suppose that (1) holds. Let δ be a fuzzy generalized bi-ideal, γ be a fuzzy right ideal and λ be a fuzzy left ideal of R. Let x ∈ R, this implies that there exists an element a ∈ R such that x = (xa)x. Now Thus

Intra-regular left almost rings
In this section, we characterize intra-regular LA-rings by the properties of fuzzy ( left, right, quasi-, bi-, generalized bi-) ideals.

Lemma 4.1: Every fuzzy left (resp. right) ideal of an intraregular LA-ring R is a fuzzy ideal of R.
Proof: Suppose that μ is a fuzzy right ideal of R. Let x, y ∈ R, this implies that there exist elements a i , b i ∈ R, Hence μ is a fuzzy ideal of R.

Lemma 4.2:
Let R be an intra-regular LA-ring with left identity e. Then every fuzzy ideal of R is a fuzzy idempotent.
Proof: Assume that μ is a fuzzy ideal of R and μ • μ ⊆ μ. Let x ∈ R, this means that there exist elements

Proposition 4.1: Let μ be a fuzzy subset of an intraregular LA-ring R with left identity e. Then μ is a fuzzy ideal of R if and only if μ is a fuzzy interior ideal of R.
Proof: Suppose that μ is a fuzzy interior ideal of R. Let x, y ∈ R, then there exist elements a i , b i ∈ R, such that So μ is a fuzzy right ideal of R, hence μ is a fuzzy ideal of R by Lemma 4.1. Converse is true by Lemma 2.11.

Remark 4.1:
The concept of fuzzy (interior, two-sided) ideals coincides in intra-regular LA-rings with left identity.

Lemma 4.3:
Let R be an intra-regular LA-ring with left identity e. Then γ ∩ μ ⊆ μ • γ , for every fuzzy left ideal μ and for every fuzzy right ideal γ of R.
Proof: Let μ be a fuzzy left ideal and γ be a fuzzy right ideal of R. Let x ∈ R, this implies that there exist (1) R is an intra-regular.
(2) μ ∩ γ = (μ • γ ) • μ for every fuzzy quasi-ideal μ and for every fuzzy ideal γ of R. Proof: Suppose that (1) holds. Let δ be a fuzzy generalized bi-ideal and γ be a fuzzy ideal of R. Now Thus (1) implies (4). It is clear that (4) ⇒ (3) and (3) ⇒ (2). Assume that (2) is true. Let μ be a fuzzy right ideal and γ be a fuzzy twosided ideal of R. Since every fuzzy right ideal of R is a fuzzy quasi-ideal of R by Lemma 2.14, so μ is a fuzzy quasi-ideal of R. By our assumption Therefore R is an intra-regular by Theorem 4.1, i.e. (1) R is an intra-regular.
(2) μ ∩ γ ⊆ γ • μ for every fuzzy quasi-ideal μ and for every fuzzy left ideal γ of R. Proof: Assume that (1) holds. Let δ be a fuzzy generalized bi-ideal and γ be a fuzzy left ideal of R. Let x ∈ R, this means that there exist elements Hence (1) implies (4). It is clear that (4) ⇒ (3) and (3) ⇒ (2). Suppose that (2) holds. Let μ be a fuzzy right ideal and γ be a fuzzy left ideal of R. Since every fuzzy right ideal of R is a fuzzy quasi-ideal of R, this implies that μ is a fuzzy quasi-ideal of R. By our supposition, μ ∩ γ ⊆ γ • μ. Thus R is an intra-regular by Theorem 4.1, i.e. (1) R is an intra-regular.

Regular and intra-regular left almost rings
In this section, we characterize both regular and intraregular LA-rings by the properties of fuzzy (left, right, quasi-, bi-, generalized bi-) ideals. (1) R is a both regular and an intra-regular.
for all fuzzy biideals μ 1 and μ 2 of R.
Proof: Suppose that (1) holds. Let μ be a fuzzy bi-ideal of R and μ • μ ⊆ μ. Let x ∈ R, this implies that there exists an element a ∈ R such that x = (xa)x, also there exist elements Hence μ = μ • μ, i.e. (1) implies (2). Assume that (2) is true. Let μ 1 and μ 2 be two fuzzy bi-ideals of R, then μ 1 ∩ μ 2 and μ 1 • μ 2 be also fuzzy bi-ideals of R. By our assumption μ 1 Again by our supposition (3) holds. Let μ be a fuzzy right ideal and γ be a fuzzy ideal of R, then μ and γ be also fuzzy bi-ideals of R, because every fuzzy right ideal and fuzzy ideal of R is a fuzzy bi-ideal of R. By our supposition Hence R is both a regular and an intra-regular, i.e. (1) R is both regular and intra-regular.
(2) Every fuzzy quasi-ideal of R is a fuzzy idempotent.
Proof: Suppose that R is both a regular and an intraregular. Let μ be a fuzzy quasi-ideal of R. Then μ be a fuzzy bi-ideal of R and μ • μ ⊆ μ. Let x ∈ R, this means that there exists an element a ∈ R such that x = (xa)x, and also there exist elements Hence μ = μ • μ. Conversely, assume that every fuzzy quasi-ideal of R is a fuzzy idempotent. Let a ∈ R, then Ra is a left ideal of R containing a by Lemma 3.5.This implies that Ra is a quasi-ideal of R, so χ Ra is a fuzzy quasi-ideal of R by Theorem 2.4. By our assumption χ Ra = χ Ra • χ Ra = χ (Ra)(Ra) , i.e. Ra = (Ra)(Ra). Since a ∈ Ra, i.e. a ∈ (Ra)(Ra). Thus a is both a regular and an intra-regular by Theorems 3.1 and 4.1, respectively. Hence R is both a regular and an intra-regular. (1) R is both regular and intra-regular.
(1) R is both regular and intra-regular.

Conclusion
In this paper, we present a brief study of the existing concept of non-associative rings with fuzzy notion and explore various properties this algebraic structure means to non-associative rings (LA-ring). This study explains the voluminous work in different fields of nonassociative rings and through which various algebraic structures in theoretical point of view could be developed. We hope that this work will provide an endless source of inspiration for future research in nonassociative ring theory.