Redefined generalized fuzzy ideals of near-rings

Appl. Math. J. Chinese Univ. 2010, 25(3): 341-348

Redefined generalized fuzzy ideals of near-rings ZHAN Jian-ming1

YIN Yun-qiang2

Abstract. With a new idea, we redefine generalized fuzzy subnear-rings (ideals) of a nearring and investigate some of its related properties. Some new characterizations are given. In particular, we introduce the concepts of strong prime (or semiprime) (∈, ∈ ∨ q)-fuzzy ideals of near-rings, and discuss the relationship between strong prime (or semiprime) (∈, ∈ ∨ q)-fuzzy ideals and prime (or semiprime) (∈, ∈ ∨ q)-fuzzy ideals of near-rings.

§1

Introduction

A near-ring satisfies all the axioms of an associative ring expect for commutativity of addition and one of the two distributive laws. After the introduction of fuzzy sets by Zadeh, there have been a number of generalizations of this fundamental concept. Abou-Zaid [1] introduced fuzzy subnear-rings (ideals) and studied some of their related properties in near-rings. See [4,5,8,9] for more work in this area. A new type of fuzzy subgroup, the (∈, ∈ ∨ q)-fuzzy subgroup, was introduced in an earlier paper of Bhakat and Das [2] by combining “belongingness” and “quasicoincidence” of fuzzy points and fuzzy sets. In fact, (∈, ∈ ∨ q)-fuzzy subgroup is an important generalization of ordinary fuzzy subgroup. It is now natural to investigate similar type of generalizations of the existing fuzzy subsystems with other algebraic structures, see [3,6,7]. In [3], Davvaz introduced (∈, ∈ ∨ q)-fuzzy subnear-rings (ideals) of near-rings and investigated some of their related properties. One of the present authors [7] considered (∈, ∈ ∨q)-fuzzy subnear-rings (ideals) of near-rings and obtained some of their related properties. The relationships among ordinary fuzzy subnear-rings (ideals), (∈, ∈ ∨ q)-fuzzy subnear-rings (ideals), and (∈, ∈ ∨ q)-fuzzy subnear-rings (ideals) of near-rings were investigated. Some characterizations of near-rings by means of (∈, ∈ ∨ q)-fuzzy ideals were also given. Received: 2009-04-06. MR Subject Classification: 16Y30, 16Y99. Keywords: Near-ring, subnear-ring (ideal), (∈, ∈∨q)-fuzzy subnear-ring (ideal), prime (semiprime) (∈, ∈ ∨ q)fuzzy subnear-ring (ideal). Digital Object Identifier(DOI): 10.1007/s11766-010-2245-6. Supported by the National Natural Science Foundation of China (60875034), the Natural Science Foundation of Education Committee of Hubei Province (D20092901), and the Natural Science Foundation of Hubei Province (2009CDB340).

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Based on [3,7], we continue to study this topic. With a new idea, we redefine generalized fuzzy subnear-rings (ideals) of a near-ring and investigate some of their related properties. We also give some new characterizations. We introduce strong prime (or semiprime) (∈, ∈ ∨ q)fuzzy ideals of near-rings, and discuss the relationship between strong prime (or semiprime) (∈, ∈ ∨ q)-fuzzy ideals and prime (or semiprime) (∈, ∈ ∨ q)-fuzzy ideals in near-rings.

§2

Preliminaries

A non-empty set R with two binary operation “ + ” and “ · ” is called a left near-ring if it satisfies the following conditions: (1) (R, +) is a group. (2) (R, ·) is a semigroup. (3) x · (y + z) = x · y + x · z for all x, y, z ∈ R. By a near-ring we always mean a left near-ring, and we write xy for x · y. An ideal I of a near-ring R is a subset of R such that (i) (I, +) is a normal subgroup of (R, +), (ii) RI ⊆ I, (iii) (x + a)y − xy ∈ I for any a ∈ I and x, y ∈ R. An ideal P of R is prime if IJ ⊆ P implies I ⊆ P or J ⊆ P for all ideals I and J of R. An ideal P of R is semiprime if I 2 ⊆ P implies I ⊆ P for all ideals I of R. In what follows, R is always a near-ring unless otherwise specified. Recall that a fuzzy set is a function μ : R → [0, 1]. We define μ−1 by μ−1 (x) = μ(−x) for all x ∈ R. For any A ⊆ R, the characteristic function of A is denoted by χA . For any t ∈ (0, 1], define a fuzzy set tA of R by t if x ∈ A, tA (x) = 0 if x ∈ A for all x ∈ R. A fuzzy set μ of R of the form

t(= 0) if y = x, 0 if y = x is said to be a fuzzy point with support x and value t and is denoted by xt . A fuzzy point xt is said to belong to (or quasi-coincident with) a fuzzy set μ, written as xt ∈ μ (or xt qμ) if μ(x) ≥ t (or μ(x) + t > 1). If xt ∈ μ or xt qμ, then we write xt ∈ ∨ qμ. If μ(x) < t (or μ(x) + t ≤ 1), then we write xt ∈μ (or xt qμ). We write ∈ ∨ q to mean that ∈ ∨ q does not hold. μ(y) =

Definition 2.1. [1] A fuzzy set μ of R is called a fuzzy subnear-ring of R if for all x, y, a ∈ R, the following conditions are satisfied: (F1a) μ(x + y) ≥ μ(x) ∧ μ(y), (F1a’) μ(−x) ≥ μ(−x), (F1b) μ(xy) ≥ μ(x) ∧ μ(y). Moreover, μ is called a fuzzy ideal of R if μ is a fuzzy subnear-ring of R and

ZHAN Jian-ming, YIN Yun-qiang.

Redefined generalized fuzzy ideals of near-rings

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(F1c) μ(y + x − y) ≥ μ(x), (F1d) μ(xy) ≥ μ(y), (F1e) μ((x + a)y − xy) ≥ μ(x). Definition 2.2. [3] A fuzzy set μ of R is called an (∈, ∈ ∨ q)-fuzzy subnear-ring of R if for all t, r ∈ (0, 1] and x, y, a ∈ R, the following conditions are satisfied: (F2a) xt ∈ μ and yr ∈ μ imply (x + y)t∧r ∈ ∨ qμ, (F2a’) xt ∈ μ implies (−x)t ∈ ∨ qμ, (F2b) xt ∈ μ and yr ∈ μ imply (xy)t∧r ∈ ∨ qμ. Moreover, μ is called an (∈, ∈ ∨ q)-fuzzy ideal of R if μ is an (∈, ∈ ∨ q)-fuzzy subnear-ring of R and (F2c) xt ∈ μ implies (y + x − y)t ∈ ∨ qμ, (F2d) yr ∈ μ and x ∈ R imply (xy)r ∈ ∨ qμ, (F2e) at ∈ μ implies ((x + a)y − xy)t∧r ∈ ∨ qμ. Theorem 2.1. [3] A fuzzy set μ of R is an (∈, ∈ ∨ q)-fuzzy subnear-ring of R if and only if for any x, y, a ∈ R, (F3a) μ(x + y) ≥ μ(x) ∧ μ(y) ∧ 0.5, (F3a’) μ(−x) ≥ μ(x) ∧ 0.5, (F3b) μ(xy) ≥ μ(x) ∧ μ(y) ∧ 0.5. Moreover, μ is an (∈, ∈ ∨ q)-fuzzy ideal of R if μ satisfies the above conditions and (F3c) μ(y + x − y) ≥ μ(x) ∧ 0.5, (F3d) μ(xy) ≥ μ(y) ∧ 0.5, (F3e) μ((x + a)y − xy) ≥ μ(a) ∧ 0.5. Definition 2.3. [7] A fuzzy set μ of R is called an (∈, ∈ ∨ q)-fuzzy subnear-ring of R if for all t, r ∈ (0, 1] and for all x, y, a ∈ R, (F4a) (x + y)t∧r ∈μ implies xt ∈ ∨ qμ or yr ∈ ∨ qμ, (F4a’) (−x)t ∈μ implies (−x)t ∈ ∨ qμ, (F4b) (xy)t∧r ∈μ implies xt ∈ ∨ qμ or yr ∈ ∨ qμ. Moreover, μ is called an (∈, ∈ ∨ q)-fuzzy ideal of R if μ is an (∈, ∈ ∨ q)-fuzzy subnear-ring of R and (F4c) (y + x − y)t ∈μ implies xt ∈ ∨ qμ, (F4d) (xy)r ∈μ and x ∈ R imply yr ∈ ∨ qμ , (F4e) ((x + a)y − xy)t∧r ∈μ implies at ∈ ∨ qμ. Theorem 2.2. [7] A fuzzy set μ of R is an (∈, ∈ ∨ q)-fuzzy subnear-ring of R if and only if for any x, y, a ∈ R, (F5a) μ(x + y) ∨ 0.5 ≥ μ(x) ∧ μ(y), (F5a’) μ(−x) ∨ 0.5 ≥ μ(x), (F5b) μ(xy) ∨ 0.5 ≥ μ(x) ∧ μ(y). Moreover, μ is an (∈, ∈ ∨ q)-fuzzy ideal of R if μ satisfies the above conditions and (F5c) μ(y + x − y) ∨ 0.5 ≥ μ(x),

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(F5d) μ(xy) ∨ 0.5 ≥ μ(y), (F5e) μ((x + a)y − xy) ∨ 0.5 ≥ μ(a). Definition 2.4. Let μ and ν be fuzzy sets of R. The product of μ and ν is defined by (μ(a) ∧ ν(b)) (μ ◦ ν)(x) = x=ab

and (μ ◦ ν)(x) = 0 if x cannot be expressed as x = ab. Definition 2.5. Let μ and ν be fuzzy sets of R. Then the sum of μ and ν is defined by (μ(a) ∧ ν(b)) (μ + ν)(x) = x=a+b

and (μ + ν)(x) = 0 if x cannot be expressed as x = a + b. In particular, for any fuzzy set μ of R and any element x of R, the sum of x and μ is given by (x + μ)(y) = μ(a) y=x+a

and (x + μ)(y) = 0 if y cannot be expressed as y = x + a.

§3

Redefined generalized fuzzy subnear-rings (ideals)

Let μ and ν be any two fuzzy sets of R. If xt ∈ μ implies xt ∈ ∨ qν for all x ∈ R and t ∈ (0, 1], then we write μ ⊆ ∨qν. If xt ∈μ implies xt ∈ ∨ qν for all x ∈ R and t ∈ (0, 1], then we write μ ⊇ ∨qν. Proposition 3.1. For any two fuzzy sets μ and ν of R, (i) μ ⊆ ∨qν if and only if ν(x) ≥ min{μ(x), 0.5} for all x ∈ R; (ii) μ ⊇ ∨qν if and only if max{μ(x), 0.5} ≥ ν(x) for all x ∈ R. Proof. (i) Let μ ⊆ ∨qν. If there exists x ∈ R such that ν(x) < t = μ(x) ∧ 0.5, then xt ∈ μ. But xt ∈ ∨ qν, a contradiction. Conversely, let ν(x) ≥ μ(x) ∧ 0.5 for all x ∈ R. If μ⊆ ∨qν, then there exists xt ∈ μ with xt ∈ ∨ qν, and so μ(x) ≥ t, ν(x) < t, and ν(x) < 0.5, a contradiction. (ii) Let μ ⊇ ∨qν. If there exists x ∈ R such that μ(x) ∨ 0.5 < t = ν(x), then xt ∈ ν with xt ∈μ and t > 0.5. Hence xt ∈ ∨ qν, and so xt qν. Therefore ν(x) + t ≤ 1, and so t ≤ 0.5, a contradiction. Conversely, let μ(x) ∨ 0.5 ≥ ν(x) for all x ∈ R. If μ⊇ ∨ qν, then there exists xt ∈μ but xt ∈ ∨ qν. Hence μ(x) < t, ν(x) ≥ t and ν(x) + t > 1. Case 1. If μ(x) > 0.5, then μ(x) ≥ ν(x), a contradiction. Case 2. If μ(x) ≤ 0.5, then 0.5 ≥ ν(x). Since ν(x) ≥ t and 0.5 ≥ ν(x) ≥ t, we have 2ν(x) ≥ ν(x) + t > 1, and so ν(x) > 0.5, a contradiction. Definition 3.1. A fuzzy set μ of R is called a new (∈, ∈ ∨ q)-fuzzy subnear-ring of R if it satisfies: (F6a) (μ + μ) ⊆ ∨qμ,



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(F6a’) μ−1 ⊆ ∨qμ, (F6b) (μ ◦ μ) ⊆ ∨qμ. Moreover, μ is called a new (∈, ∈ ∨ q)-fuzzy ideal of R if it is a new (∈, ∈ ∨ q)-fuzzy subnearring of R and if for all x, y ∈ R, (F6c) (x + μ − x) ⊆ ∨qμ, (F6d) (χR ◦ μ) ⊆ ∨qμ, (F6e) ((x + μ) ◦ y − xy) ⊆ ∨qμ. Theorem 3.1. (1) A fuzzy set μ of R is a new (∈, ∈ ∨ q)-fuzzy subnear-ring of R if and only if it satisfies (F3a), (F3a’) and (F3b). (2) A fuzzy set μ of R is a new (∈, ∈ ∨ q)-fuzzy ideal of R if and only if it is a new (∈, ∈ ∨ q)fuzzy subnear-ring of R and satisfies (F3c), (F3d) and (F3e). Proof. (1) Let μ be a new (∈, ∈ ∨ q)-fuzzy subnear-ring of R. If there exist x, y ∈ R such that μ(x + y) < t < μ(x) ∧ μ(y) ∧ 0.5, then t < 0.5, xt ∈ μ, yt ∈ μ, and (x + y)t ∈ ∨ qμ. Since (μ(a) ∧ μ(b)) ≥ μ(x) ∧ μ(y) ≥ t, (μ + μ)(x + y) = x+y=a+b

we have (x + y)t ∈ (μ + μ). Thus (x + y)t ∈ ∨ qμ, a contradiction. This proves that (F3a) holds. If there exists x ∈ R such that μ(−x) < t < μ(x) ∧ 0.5, then μ(x) ≥ t, t < 0.5 and (−x)t ∈ ∨ qμ. But μ−1 (−x) = μ(x) ≥ t, so (−x)t ∈ μ−1 , which implies (−x)t ∈ ∨ qμ, a contradiction. This proves that (F3a’) holds. The proof of (F3b) is similar to the proof of (F3a). Conversely, assume that μ satisfies (F3a), (F3a’) and (F3b). Let xt ∈ (μ + μ) but xt ∈ ∨ qμ. Then μ(x) < t and μ(x) < 0.5. By definition, (μ + μ)(x) = (μ(a) ∧ μ(b)). Since 0.5 > x=a+b

μ(x) = μ(a + b) ≥ μ(a) ∧ μ(b) ∧ 0.5, and so μ(x) ≥ μ(a) ∧ μ(b). Thus t ≤ (μ + μ)(x) ≤ μ(x) = μ(x), x=a+b

that is, μ(x) ≥ t, a contradiction. This proves that (F6a) holds. Now let xt ∈ μ−1 but xt ∈ ∨ qμ. Then μ(x) < t and μ(x) < 0.5. Thus μ(−x) ≥ μ(x) ∧ 0.5 = μ(x), and so μ(−x) = μ(x). Hence t ≤ μ−1 (x) = μ(−x) = μ(x), a contradiction. The proof of (F6b) is similar to the proof of (F6a). Therefore, μ is a new (∈, ∈ ∨ q)-fuzzy subnear-ring of R. (2) Let μ be a new (∈, ∈ ∨ q)-fuzzy ideal of R. We only prove that μ satisfies (F3c). The others are similar. If there exist x, y ∈ R such that μ(y + x − y) < t < μ(x) ∧ μ(y) ∧ 0.5, then t < 0.5, xt ∈ μ, and (y + x − y)t ∈ ∨ qμ. Since μ(a) ≥ μ(x) ≥ t, (y + μ − y)(y + x − y) = y+x−y=y+a−y

we have (y + x − y)t ∈ (y + μ − y). Thus (y + x − y)t ∈ ∨ qμ, a contradiction. This proves that (F3a) holds. Conversely, assume that the conditions hold. We only prove (F6c) holds. The others are similar. Let x, y ∈ R and t ∈ (0, 1] be such that xt ∈ (y + μ − y) but xt ∈ ∨ qμ. Then μ(x) < t μ(a). Since 0.5 > μ(x) = μ(y + a − y) ≥ and μ(x) < 0.5. By definition, (y + μ − y)(x) = x=y+a−y

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μ(a) ∧ 0.5, so μ(x) ≥ μ(a). Thus, t ≤ (y + μ − y)(x) ≤

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μ(x) = μ(x), that is, μ(x) ≥ t,

a contradiction. This proves that (F6c) holds. Therefore, μ is a new (∈, ∈ ∨ q)-fuzzy ideal of R. The following is a consequence of Theorems 2.1 and 3.1. Corollary 3.1. The concepts of new (∈, ∈ ∨ q)-fuzzy subnear-rings (ideals) of R and (∈, ∈ ∨ q)fuzzy subnear-rings (ideals) are equivalent. We now consider another generalized fuzzy subnear-ring (or ideal) of near-ring. Definition 3.2. A fuzzy set μ of R is called a new (∈, ∈ ∨ q)-fuzzy subnear-ring of R if it satisfies: (F7a) μ ⊇ ∨ q(μ + μ), (F7a’) μ ⊇ ∨ qμ−1 , (F7b) μ ⊇ ∨ q(μ ◦ μ). Moreover, μ is called a new (∈, ∈ ∨ q)-fuzzy ideal of R if it is a new (∈, ∈ ∨ q)-fuzzy subnearring of R and if for all x, y ∈ R, (F7c) μ ⊇ ∨ q(x + μ − x), (F7d) μ ⊇ ∨ q(χR ◦ μ), (F7e) μ ⊇ ∨ q((x + μ) ◦ y − xy). Theorem 3.2. (1) A fuzzy set μ of R is a new (∈, ∈ ∨ q)-fuzzy subnear-ring of R if and only if it satisfies (F5a), (F5a’) and (F5b). (2) μ is a new (∈, ∈∨q)-fuzzy ideal of R if and only if it is a new (∈, ∈∨q)-fuzzy subnear-ring of R and satisfies (F5c), (F5d) and (F5e). Proof. We only prove (1). The proof of (2) is similar. Let μ be a new (∈, ∈ ∨ q)-fuzzy subnear-ring of R. If there exist x, y ∈ R such that μ(x + y) ∨ 0.5 < t < μ(x) ∧ μ(y), then t > 0.5, xt ∈ μ, yt ∈ μ, but (x + y)t ∈μ. So (x + y)t ∈ ∨ q(μ + μ). Thus (μ + μ)(x + y) < t and (μ + μ)(x + y) + t ≤ 1. But (μ(a) ∧ μ(b)) ≥ μ(x) ∧ μ(y) ≥ t, (μ + μ)(x + y) = x+y=a+b

which implies (μ + μ)(x + y) ≥ t. Thus t ≤ 0.5, a contradiction. This proves (F5a) holds. We can prove (F5a’) and (F5b) similarly. Conversely, assume that μ satisfies (F5a), (F5a’) and (F5b). Let xt ∈ μ but xt ∈ ∨ q(μ + μ). Then μ(x) < t, but (μ + μ)(x) ≥ t and (μ + μ)(x) + t > 1. So (μ + μ)(x) > 0.5. By definition, (μ(a) ∧ μ(b)). Since 0.5 ∨ μ(x) = μ(a + b) ∨ 0.5 ≥ μ(a) ∧ μ(b), we have (μ + μ)(x) = x=a+b 0.5 ∨ μ(x) ≥ μ(a) ∧ μ(b). Thus t ≤ (μ + μ)(x) ≤ μ(x) ∨ 0.5. Since (μ + μ)(x) > 0.5, we x=a+b

have μ(x) ≥ 0.5, and so μ(x) ≥ t, a contradiction. This proves that (F7a) holds. The proofs of (F7a’) and (F7b) are similar. Therefore, μ is a new (∈, ∈ ∨ q)-fuzzy subnear-ring of R. The following is a consequence of Theorems 2.2 and 3.2.



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Corollary 3.2. The concepts of new (∈, ∈ ∨ q)-fuzzy subnear-rings (ideals) of R and (∈, ∈ ∨ q)fuzzy subnear-rings (ideals) are equivalent.

§4

Strong prime (semiprime) (∈, ∈ ∨ q)-fuzzy ideals

In this section, we define strong prime (semiprime) (∈, ∈ ∨ q)-fuzzy ideals of near-rings. We also discuss the relationship between strong prime (or semiprime) (∈, ∈ ∨ q)-fuzzy ideals and prime (or semiprime) (∈, ∈ ∨ q)-fuzzy ideals of near-rings. Definition 4.1. [7] An (∈, ∈ ∨ q)-fuzzy ideal μ of R is prime if for all x, y ∈ R and t ∈ (0, 1], we have (P) (xy)t ∈ μ ⇒ xt ∈ ∨ qμ or yt ∈ ∨ qμ. An (∈, ∈ ∨ q)-fuzzy ideal μ of R is semiprime if for all x ∈ R and t ∈ (0, 1], we have (SP) (x2 )t ∈ μ ⇒ xt ∈ ∨ qμ. Theorem 4.1. [7]An (∈, ∈ ∨ q)-fuzzy ideal μ of R is prime if for all x, y ∈ R, (P’) μ(x) ∨ μ(y) ≥ μ(xy) ∧ 0.5. An (∈, ∈ ∨ q)-fuzzy ideal μ of R is semiprime if for all x ∈ R, (SP’) μ(x) ≥ μ(x2 ) ∧ 0.5. For a fuzzy set μ of R and t ∈ (0, 1], the crisp set μt = {x ∈ R | μ(x) ≥ t} is called the level subset of μ. Theorem 4.2. [7] An (∈, ∈ ∨ q)-fuzzy ideal μ of R is prime (semiprime) if and only if μt = ∅ is a prime (semiprime) ideal of R for all t ∈ (0, 0.5]. Definition 4.2. An (∈, ∈ ∨ q)-fuzzy ideal ρ of R is strong prime if for every (∈, ∈ ∨ q)-fuzzy ideals μ and ν of R, (P”) μ ◦ ν ⊆ ρ implies μ ⊆ ρ or ν ⊆ ρ. An (∈, ∈ ∨ q)-fuzzy ideal μ of R is strong semiprime if for every (∈, ∈ ∨ q)-fuzzy ideal μ of R, (SP”) μ ◦ μ ⊆ ρ implies μ ⊆ ρ. Theorem 4.3. Let μ be a strong prime (semiprime) (∈, ∈ ∨ q)-fuzzy ideal of R. Then μt = ∅ is a prime (semiprime) ideal of R for all t ∈ (0, 0.5]. Proof. We only consider strong prime (∈, ∈ ∨ q)-fuzzy ideals. The case for strong semiprime ideal is similar. Let t ∈ (0, 0.5] be such that μt is non-empty. Then μt is an ideal of R. Now we show that μt is prime. Let I and J be two ideals of R such that IJ ⊆ μt . Then it is easy to see that tI and tJ are two (∈, ∈ ∨ q)-fuzzy ideals of R and that tI ◦ tJ ⊆ μ. For x ∈ R, if (tI ◦ tJ )(x) = 0, then (tI ◦ tJ )(x) = 0 ≤ μ(x). Otherwise, there exist a, b ∈ R such that x = ab and tI (a) ∧ tJ (b) = 0. This implies that a ∈ I and b ∈ J. Hence x ∈ IJ ⊆ μt , that is, μ(x) ≥ t. Hence (tI ◦ tJ )(x) = tI (y) ∧ tJ (z) ≤ t ≤ μ(x). Therefore, tI ◦ tJ ⊆ μ. Since μ is a strong x=yz

prime (∈, ∈ ∨ q)-fuzzy ideal of R, we have tI ⊆ μ or tJ ⊆ μ. This implies that I ⊆ μt or J ⊆ μt .

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The following is a consequence of Theorems 4.2 and 4.3. Theorem 4.4. Every strong prime (semiprime) (∈, ∈ ∨ q)-fuzzy ideal of a near-ring is a prime (semiprime) (∈, ∈ ∨ q)-fuzzy ideal. Remark 4.1. The converse of Theorem 4.4 is not true in general as shown by the following example. Let (Z, +, ·) be the near-ring (also a ring) of all integers. Then 0.4(2) is an (∈, ∈ ∨ q)fuzzy ideal of Z and the non-empty subset (0.4(2) )t is a prime ideal of R for all t ∈ (0, 0.5]. By Theorem 4.2, we know that 0.4(2) is a prime (∈, ∈ ∨ q)-fuzzy ideal of Z, but it is not strong prime. In fact, 0.4(3) ◦ 0.5(4) ⊆ 0.4(2) , but 0.4(3) 0.4(2) and 0.5(4) 0.4(2) , in which both 0.4(3) and 0.5(4) are (∈, ∈ ∨ q)-fuzzy ideals of Z.

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Department of Mathematics, Hubei Institute for Nationalities, Enshi 445000, China Email: [email protected] College of Mathematics and Information Sciences, East China Institute of Technology, Fuzhou, Jiangxi 344000, China