A study of fuzzy prime near-rings involving fuzzy

semigroup ideals

M. OU-MHA1, A. RAJI1, M. OUKESSOU1, A. BOUA2

1LMACS Laboratory, Faculty of Sciences and Technology

Sultan Moulay Slimane University, Beni Mellal,

MOROCCO

2Department of Mathematics, Polydisciplinary Faculty of Taza

Sidi Mohammed Ben Abdellah University, Fez,

MOROCCO

Abstract: In this paper, our main objective is to introduce the notion of a fuzzy semigroup ideal by using

Yuan and Lee’s deﬁnition of fuzzy group based on fuzzy binary operations

Also, some of its basic

properties are studied analogously to the known results in the case of semigroup ideals deﬁned in the

framework of ordinary near-rings.

Key-Words: Fuzzy group, Fuzzy near-rings, Semigroup ideals, Prime near-rings, Binary operations,

Commutativity.

Received: August 27, 2023. Revised: April 6, 2024. Accepted: April 28, 2024. Published: May 21, 2024.

1 Introduction

In [1], gave the deﬁnition of the fuzzy binary op-

eration using the notion of a fuzzy subset of a

fuzzy set introduced by Zadeh in his famous pa-

per, [2], published in 1965. Taking advantage of

this deﬁnition, extensive work has been published

by several researchers (see, [3], [4], [5], for further

references). In 2004, a new deﬁnition of fuzzy

group was created by the two researchers, [6],

they also presented the notion of commutativity

of a fuzzy group and some of its basic principles.

After these studies, in [7], created the concept

of fuzzy ring based on the deﬁnition of [6], of a

fuzzy group, and they obtained interesting results

on this subject. Motivated by the classical theory

of near-rings, we refer the reader to [8], and the

work of [7], in which the two operations ∗and ◦

are two mappings constructed from the fuzzy bi-

nary operations Tand Las given in the section

of preliminary, we succeed to deﬁne the notion of

fuzzy near-ring as follows:

Deﬁnition 1. For any nonempty set Xwith two

fuzzy binary operations Tand Lis said fuzzy left

near-ring if the following assertions hold:

i) (X, T )is a fuzzy group not necessarily commu-

tative,

ii)∀a, b, c, x1,x2∈X, we have (a∗(b∗c))(x1)> θ

and ((a∗b)∗c)(x2)> θ =⇒x1=x2,

iii)∀a, b, c, x1, x2∈X, we have

(a∗(b◦c))(x1)> θ and ((a∗b)◦(a∗c))(x2)> θ

=⇒x1=x2,

where θ∈[0,1) is a ﬁxed number. Further, if we

replace the last condition by:

iv)∀a, b, c, x1, x2∈X, (b◦c)∗a)(x1)> θ and

((b∗a)◦(c∗a))(x2)> θ =⇒x1=x2,

then (X, T, L)is called right fuzzy near-ring.

And (X, T, L)is said to be a fuzzy ring, when

(X, T, L)is a left and right fuzzy near-ring and

(X, T )is abelain. Moreover, according to, [7, Def-

inition 9],(X, T, L)is said to be a commutative

fuzzy ring if (a∗b)(u)> θ ⇔(b∗a)(u)> θ for

all a, b ∈X.

Noting that (X, T, L) is called a prime fuzzy

near-ring, if it has the property that ((x∗y)∗

z)(e)> θ for all x, y, z ∈Ximplies that x=e

or z=e. Also, ZF(X) = {x∈X / L(x, y, z)>

θ⇐⇒ L(y, x, z)> θ, ∀y, z ∈X}denote the

fuzzy multiplicative center of X.

2 Preliminary results

In this section, we will formulate some basic

deﬁnitions and results that will be essential for

the rest of this paper.

Deﬁnition 2. [9, Deﬁnition 2.1] Let Xbe a

nonempty set and Tbe a fuzzy subset of X×X×X

and θ∈[0,1) is a ﬁxed number. Tis called a

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fuzzy binary operation on Xif the following con-

ditions hold:

(C1)∀x, y ∈X, ∃z∈Xsuch that T(x, y, z)> θ.

(C2)∀x, y, t1, t2∈X, T (x, y, t1)> θ and

T(x, y, t2)> θ implies t1=t2.

Let Tand Lbe two fuzzy binary operations on

X, then we can deﬁned the following mappings:

◦:F(X)×F(X)−→ F(X)

(µ, v)7−→ µ◦v

and

∗:F(X)×F(X)−→ F(X)

(µ, v)7−→ µ∗v,

where F(X) = {µ|µ:X−→ [0,1]},and for all

µ, v ∈F(X),we have







(µ∗v)(z) = W

x,y ∈Xµ(x)∧v(y)∧L(x, y, z),

(µ◦v)(z) = W

x,y ∈Xµ(x)∧v(y)∧T(x, y, z).

Let x, y ∈X,µ={x}and v={y}, and let µ◦v

and µ∗vbe denoted by x◦yand x∗y, respectively.

Then, we have for all z, t ∈X

(x◦y)(z) = T(x, y, z),(1)

(x∗y)(z) = L(x, y, z),(2)

((x◦y)◦z)(t) = W

h∈XT(x, y, h)∧T(h, z, t),(3)

(x◦(y◦z))(t) = W

h∈XT(y, z, h)∧T(x, h, t),(4)

((x∗y)∗z)(t) = W

h∈XL(x, y, h)∧L(h, z, t),(5)

(x∗(y∗z))(t) = W

h∈XL(y, z, h)∧L(x, h, t),(6)

(x∗(y◦z))(t) = W

h∈XT(y, z, h)∧L(x, h, t),(7)

((x∗y)◦(x∗z))(t) =

d,h∈XL(x, y, d)∧L(x, z, h)∧T(d, h, t).

(8)

Deﬁnition 3. [9, Deﬁnition 2.2] Let Xbe a

nonempty set and Ta fuzzy binary operation on

X. Then (X, T )is called a fuzzy group if the fol-

lowing conditions hold:

1. ∀a, b, c, c1, c2∈X, ((a◦b)◦c)(c1)> θ and

(a◦(b◦c))(c2)> θ =⇒c1=c2.

2. ∃e∈Xsuch that for all x∈X,(e◦x)(x)> θ

and (x◦e)(x)> θ.eis called the identity

element of (X, T ).

3. ∀x∈X, ∃y∈Xsuch that (xoy)(e)> θ

and (y o x)(e)> θ.yis called the inverse

element of xand denoted by x−1.

Lemma 1. [6, Proposition 2.1] Let (X, T )be a

fuzzy group, then

1) (x◦y)(a)> θ and (x◦z)(a)> θ =⇒y=z;

2) (a◦x)(y)> θ and (b◦x)(y)> θ =⇒a=b;

3) (a◦b)(c)> θ and (b−1◦a−1)(d)> θ =⇒

d=c−1;

4) (a◦a)(a)> θ =⇒a=e;

5) (a−1)−1=a.

Deﬁnition 4. [7, Deﬁnition 6] Let (X, T )be a

fuzzy group. (X, T )is called abelian fuzzy group

if we have, for all x, y, z ∈X,

T(x, y, z)> θ ⇐⇒ T(y, x, z)> θ.

Lemma 2. [9, Theorem 3.1 & Theorem 3.3]

1- Let (X, T, L)be a left fuzzy near-ring, then

ZF(T) =

{x∈X| ∀y∈X, ((x∗y)◦(y∗x−1))(e)> θ}.

2- Let (X, T, L)be a right fuzzy near-ring, then

ZF(T) =

{x∈X| ∀y∈X, ((x∗y)◦(y−1∗x))(e)> θ}.

Lemma 3. [9, Proposition 3.1 & Proposition

3.2]

1- Let (X, T, L)be a left fuzzy near-ring, then

∀x, y, z ∈X, ((x∗y)∗z)◦((x∗y)∗z−1)(e)> θ.

2- Let (X, T, L)be a right fuzzy near-ring, then

∀x, y, z ∈X, (x∗(y∗z))◦(x−1∗(y∗z))(e)> θ.

Lemma 4. [9, lemma 3.1 & lemma 3.2]

1- Let (X, T, L)be a left fuzzy near-ring, then

∀k, x ∈X, (k∗x)∗(k∗x−1)(e)> θ.

2- Let (X, T, L)be a right fuzzy near-ring, then

∀x, k ∈X, (k∗x)∗(k−1∗x)(e)> θ.

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3 Main results

In this section, we deﬁne the notion of fuzzy

left semigroup ideal (resp. fuzzy right semigroup

ideal) of a fuzzy near-ring and we prove some

of their basic properties which are analogous to

those of the classical semigroup theory in the case

of near-rings. Also, we show that under some con-

ditions on fuzzy left semigroup (resp. fuzzy right

semigroup ideal), the fuzzy near-rings must be a

fuzzy commutative rings.

Deﬁnition 5. Let (X, T, L)be a fuzzy near-ring

and Ia nonempty subset of X, then Iis called

1. A fuzzy left semigroup ideal of Xif (x∗

s)(t)> θ implies t∈I,∀x∈X, ∀s∈I.

2. A fuzzy right semigroup ideal of Xif (s∗

x)(t)> θ implies t∈I,∀x∈X, ∀s∈I.

3. A fuzzy semigroup ideal of Xif is both right

and left fuzzy semigroup ideal. Moreover, I

is said to be non trivial if I6={e}.

Lemma 5. Let (X, T, L)be a fuzzy prime near-

ring, Ia nontrivial fuzzy semigroup ideal of X

and let x∈X.

i)If for all y∈I, (x∗y)(e)> θ then x=e,

ii)If for all y∈I, (y∗x)(e)> θ then x=e.

Proof. i) Assume that for all y∈I, (x∗y)(e)> θ.

Letting z∈X, s ∈I, then there exits t∈X

satisﬁes (z∗s)(t) = L(z, s, t)> θ. Since Iis a

fuzzy semigroup ideal of X, it follows that t∈I.

In particular, putting y=tin our assumption,

we get

L(x, t, e)=(x∗t)(e)> θ. (9)

Consequently,

x∗z∗s(e)≥L(z, s, t)∧L(x, t, e)> θ. (10)

In view of the fuzzy primeness of (X, T, L), the

last result shows that x=eor s=e. Taking

into account that Iis nontrivial, we can consider

s6=eand therefore x=e.

ii) Consider z∈Xand s∈I, there exits t∈X

such that (s∗z)(t) = L(s, z, t)> θ. Using the

same argument as used above, we infer that

s∗z∗x(e)≥L(s, z, t)∧L(t, x, e)> θ. (11)

By the fuzzy primeness of (X, T, L) and Iis non-

trivial, we get the required result.

In the case of classical near-rings, in [10],

showed in the case of a 3-prime near-ring N, if

xIy={0}then x= 0 or y= 0, where Iis a semi-

group ideal of N. The following theorem treats

this result in the case of a fuzzy semigroup ideal.

Theorem 1. Let (X, T, L)be a fuzzy prime near-

ring, Ia nontrivial fuzzy semigroup ideal of X

and let x, y ∈X. Then,

∀r∈I, x∗r∗y(e)> θ =⇒x=eor y=e.

Proof. Suppose that ∀r∈I, x∗r∗y(e)> θ.

Let (z, s)∈X×Iand taking t∈Xsatisfying

(z∗s)(t)> θ. Using the fact that Iis a fuzzy

semigroup ideal of X, we conclude that t∈Iand

hence,

x∗t)∗y(e)> θ. (12)

Let h∈Xsuch that L(x, t, h)> θ, then equation

(12) proves that L(h, y, e)> θ. Also,

x∗(z∗s)(h)≥L(z, s, t)∧L(x, t, h)> θ, (13)

Let v, l ∈Xsuch that L(x, z, v)> θ and

L(v, s, l)> θ, then

(x∗z)∗s(l)≥L(x, z, v)∧L(v, s, l)> θ. (14)

Deﬁnition 1 (ii) together (13) and (14) show that

l=h, so that L(v, s, h)> θ, which implies that

(v∗s)∗y)(e)≥L(v, s, h)∧L(h, y, e)> θ. (15)

Let kand nbe two elements of Xwhich are sat-

isfying L(s, y, k)> θ and L(v, k, n)> θ. So,

v∗(s∗y)(n)≥L(s, y, k)∧L(v, k, n)> θ. (16)

Once again Deﬁnition 1 ii) forces n=ewhich

gives L(v, k, e)> θ. Consequently,

(x∗z)∗k(e)≥L(x, z, v)∧L(v, k, e)> θ. (17)

In virtue of the fuzzy primeness of (X, T, L), the

latter result shows that x=eor k=e. Now,

suppose that k=e, then L(s, y, e)> θ which

means that (s∗y)(e)> θ. Since sis an arbitrary

element of I, then Lemma 5 (ii) assures that y=

eand consequently,

x=eor y=e.

Lemma 6. Let (X, T )be a fuzzy group. If for all

x, y ∈X, ∃t∈Xsuch that

(x◦x)◦(y◦y)(t)> θ and

(x◦y)◦(x◦y)(t)> θ,

then (X, T )is abelian.

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Proof. Let a, b, c ∈Xsuch that T(a, b, c)> θ.

Our main is to prove that T(b, a, c)> θ, for this,

choosing h∈Xwhich is satisfying T(b, a, h)> θ

and proving that h=c.

From our hypotheses, there exists t∈Xsuch

that (a◦a)◦(b◦b)(t)> θ, (18)

and (a◦b)◦(a◦b)(t)> θ. (19)

Taking t1, t2∈Xsuch that T(a, a, t1)> θ and

T(b, b, t2)> θ, thus (18) and (19) give respec-

tively T(t1, t2, t)> θ and T(c, c, t)> θ. Then,

t1◦(b◦b)(t)≥T(b, b, t2)∧T(t1, t2, t)> θ, (20)

and

c◦(a◦b)(t)≥T(a, b, c)∧T(c, c, t)> θ. (21)

Now, let x1, x2, y1, y2∈Xsatisfy T(t1, b, x1)> θ,

T(x1, b, y1)> θ,T(c, a, x2)> θ and T(x2, b, y2)>

θ. Then,

(t1◦b)◦b(y1)> θ and (c◦a)◦b(y2)> θ,

which, because of (20) and (21) together Deﬁni-

tion 3, implies that y1=tand y2=t. It follows

that T(x1, b, t)> θ and T(x2, b, t)> θ, hence

in view of Lemma 1(2) we get x1=x2. Conse-

quently, T(t1, b, x1)> θ and T(c, a, x1)> θ. So

that,

(a◦a)◦b(x1)≥T(a, a, t1)∧T(t1, b, x1)> θ,

(22)

and

(a◦b)◦a(x1)≥T(a, b, c)∧T(c, a, x1)> θ. (23)

Let v, k ∈Xsuch that T(a, c, v)> θ and

T(a, h, k)> θ. Then,

a◦(a◦b)(v)≥T(a, b, c)∧T(a, c, v)> θ

and

a◦(b◦a)(k)≥T(b, a, h)∧T(a, h, k)> θ.

(24)

Combining (22), (23), (24) and using Deﬁnition

3, we obtain v=x1and h=x1and therefore,

T(a, c, x1)> θ and T(a, h, x1)> θ. Once again,

using Lemma 1(1) we conclude that c=h. So

that, T(b, a, c)> θ.

Conversely, assuming that T(b, a, c)> θ and us-

ing similar arguments as used above to prove that

T(a, b, c)> θ. Thus, (X, T ) is a commutative

fuzzy group.

Theorem 2. Let (X, T, L)be a fuzzy near-ring.

If there exists z∈ZF(X)∗such that T(z, z, r)>

θ=⇒r∈ZF(X)for all r∈X, then (X, T )is

abelian.

Proof. Suppose that (X, T, L) is a left fuzzy near-

ring and let x, y ∈X. Consider zthe element of

our hypothesis; by Deﬁnition 2 there exists t∈X

such that T(z, z, t)> θ which, according to our

hypothesis, implies that t∈ZF(X).

Now, let v, h ∈Xsuch that T(x, y, v)> θ and

L(t, v, h)> θ. Then,

t∗(x◦y)(h)≥T(x, y, v)∧L(t, v, h)> θ. (25)

Taking h1, h2, h0∈Xsuch that L(t, x, h1)> θ,

L(t, y, h2)> θ and T(h1, h2, h0)> θ. Then,

(t∗x)◦(t∗y)(h0)≥

L(t, x, h1)∧L(t, y, h2)∧T(h1, h2, h0).

(26)

From Deﬁnition 1 (iii), (25) and (26) give h0=h,

so that

T(h1, h2, h)> θ. (27)

Also, as t∈ZF(X), we have L(x, t, h1)> θ and

L(y, t, h2)> θ, then

x∗(z◦z)(h1)≥T(z, z, t)∧L(x, t, h1)> θ and

y∗(z◦z)(h2)≥T(z, z, t)∧L(y, t, h2)> θ.

Choosing v1, v2, v3, v4∈Xsuch that L(x, z, v1)>

θ,L(y, z, v2)> θ,T(v1, v1, v3)> θ and

T(v2, v2, v4)> θ, we infer that

(x∗z)◦(x∗z)(v3)> θ and (y∗z)◦(y∗z)(v4)> θ.

(28)

Consequently, in virtue of Deﬁnition 1 (iii), we

conclude that v3=h1and v4=h2which implies

that T(v1, v1, h1)> θ and L(v2, v2, h2)> θ. Once

again, since z∈ZF(X)∗, we have L(z, x, v1)> θ

and L(z, y, v2)> θ, thus

(z∗x)◦(z∗x)(h1)> θ and (z∗y)◦(z∗y)(h2)> θ.

(29)

Next, choose `1, `2, `3, `4such that T(x, x, `1)>

θ,T(y, y, `2)> θ,L(z, `1, `3)> θ and

L(z, `2, `4)> θ, which yields

z∗(x◦x)(`3)> θ and z∗(y◦y)(`4)> θ. (30)

Invoking Deﬁnition 1 (iii), we arrive at `3=h1

and `4=h2. So that, L(z, `1, h1)> θ and

L(z, `2, h2)> θ. Because of (27), we get

(z∗`1)◦(z∗`2)(h)≥

L(z, `1, h1)∧L(z, `2, h2)∧T(h1, h2, h)> θ.

(31)

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Taking x1, h00 ∈Xsatisfy T(`1, `2, x1)> θ and

L(z, x1, h00 )> θ, we obtain

z∗(`1◦`2)(h00 )> θ, (32)

which, because of Deﬁnition 1 (iii), implies that

h=h00 and therefore,

L(z, x1, h)> θ. (33)

Similarly, since L(v, t, h)> θ, then we have

v∗(z◦z)(h)≥T(z, z, t)∧L(v, t, h)> θ. (34)

Let t1, t2∈Xsuch that L(v, z, t1)> θ and

T(t1, t1, t2)> θ, which implies that

(v∗z)◦(v∗z)(t2)≥

L(v, z, t1)∧L(v, z, t1)∧T(t1, t1, t2)> θ. (35)

Deﬁnition 1 (iii) assures that t2=hand thus,

T(t1, t1, h)> θ.

In virtue of z∈ZF(X)∗and L(v, z, t1)> θ, we

get L(z, v, t1)> θ, then (z∗v)◦(z∗v)(h)> θ.

Taking h000 , x2∈Xverify T(v, v, x2)> θ and

L(z, x2, h000 )> θ, then z∗(v◦v)(h000 )> θ which,

by Deﬁnition 1, guarantees that h000 =h, and thus

L(z, x2, h)> θ. (36)

Now, from Lemma 4 (1.), we have

(z∗x2)◦(z∗x−1

2)(e)> θ. (37)

Let m∈Xsuch that L(z, x−1

2, m)> θ and com-

bining (36) and (37), we ﬁnd that T(h, m, e)> θ.

Hence, because of (33) we get

(z∗x1)◦(z∗x−1

2)(e)≥

L(z, x1, h)∧L(z, x−1

2, m)∧T(h, m, e)> θ.

(38)

Let y1, y2∈Xsuch that T(x1, x−1

2, y1)> θ

and L(z, y1, y2)> θ. It follows that

z∗(x1◦x−1

2)(y2)≥

T(x1, x−1

2, y1)∧L(z, y1, y2)> θ.

(39)

Once again by Deﬁnition 1 (iii), (38) and (39)

shows that y2=eand thus L(z, y1, e)> θ.

Let kan arbitrary element of X, we have

k∗(z∗y1)(e)≥L(z, y1, e)∧L(k, e, e)> θ. (40)

Taking s, c ∈Xsuch that L(k, z, s)> θ and

L(s, y1, c)> θ, which proves that

(k∗z)∗y1(c)> θ. (41)

Because of Deﬁnition 1 (ii), the last two results

show that c=e, which allowed us to conclude

that L(s, y1, e)> θ.

In view of z∈ZF(X)∗and L(k, z, s)> θ, we

have L(z, k, s)> θ and hence, for all k∈X

(z∗k)∗y1(e)≥L(z, k, s)∧L(s, y1, e)> θ. (42)

In the light of the primeness of (X, T, L) and z∈

ZF(X)∗, the last relation shows that y1=e, and

thus

T(x1, x−1

2, e)> θ. (43)

Once again, from Lemma 4 (1.), we have

(z∗x−1

2)◦(z∗x2)(e)> θ. (44)

By reasoning in the same way as above, we arrive

T(x−1

2, x1, e)> θ. (45)

Now, from Deﬁnition 3 and (43) and (45), we

obtain x1= (x−1

2)−1and by lemma 1 (5), we

arrive at x1=x2. Thus,

(x◦x)◦(y◦y)(x1)≥

L(x, x, `1)∧L(y, y, `2)∧T(`1, `2, x1)> θ,

(46)

and

(x◦y)◦(x◦y)(x1)≥

T(x, y, v)∧T(x, y, v)∧T(v, v, x1)> θ.

(47)

Consequently, (X, T ) is an abelian fuzzy group by

Lemma 6. This ends the prove of our Theorem.

Remark 1. The results in the previous Theo-

rem remain valid for right fuzzy near-rings with

the obvious changes by using the second case of

Lemma 4.

Theorem 3. Let (X, T, L)be a fuzzy prime near-

ring, Ibe a nontrivial fuzzy semigroup ideal of

(X, T, L)and x∈X. If for all u∈I, there exists

t∈Xsuch that (u∗x)(t)> θ and (x∗u)(t)> θ,

then x∈ZF(X).

Proof. Suppose that (X, T, L) is a fuzzy prime left

near-ring. By Lemma 2 (1), it suﬃces to prove

that ∀y∈X, ((x∗y)◦(y∗x−1))(e)> θ. For this,

let y∈Xand u∈I.

By the deﬁnition of a fuzzy binary operation,

there are `1, `2, ` ∈Xsuch that L(x, y, `1)> θ,

L(y, x−1, `2)> θ and T(`1, `2, `)> θ which give

((x∗y)◦(y∗x−1))(`)> θ. Our goal is to show

that `=e.

Taking h∈Xsuch that (u∗y)(h) = L(u, y, h)>

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θ, in virtue of Iis a fuzzy semigroup ideal of

X, we ﬁnd that h∈I. Taking into account

our hypotheses, there exists t∈Xsatisfying

(h∗x)(t)> θ and (x∗h)(t)> θ. Then,

(u∗y)∗x(t)≥L(u, y, h)∧L(h, x, t)> θ, (48)

and

x∗(u∗y)(t)≥L(u, y, h)∧L(x, h, t)> θ. (49)

Also, by our hypotheses, there exists an element

t1∈Xsuch that (x∗u)(t1) = L(x, u, t1)> θ and

(u∗x)(t1) = L(u, x, t1)> θ. Choosing h1∈X

satisfying L(t1, y, h1)> θ. Then,

(x∗u)∗y(h1)≥L(x, u, t1)∧L(t1, y, h1)> θ.

(50)

According to the second condition of Deﬁnition 1,

the two relations (49) and (50) aﬃrm that h1=t,

then L(t1, y, t)> θ. So that,

(u∗x)∗y(t)≥L(u, x, t1)∧L(t1, y, t)> θ, (51)

Now, taking h2∈Xsatisfying L(u, `1, h2)> θ,

we get

u∗(x∗y)(h2)≥L(x, y, `1)∧L(u, `1, h2)> θ.

(52)

Once again, in view of Deﬁnition 1 (ii), (51) and

(52) give h2=tand hence L(u, `1, t)> θ. From

Lemma 3 (1), we have

(u∗y)∗x◦(u∗y)∗x−1(e)> θ. (53)

Let v, t2∈Xsuch that L(u, y, t2)> θ and

L(t2, x−1, v)> θ, which implies that

((u∗y)∗x−1)(v)≥L(u, y, t2)∧L(t2, x−1, v)> θ.

(54)

Using (48), (53) and (54), we infer that

T(t, v, e)> θ. Choose h3∈Xsuch that

L(u, `2, h3)> θ, then

u∗(y∗x−1)(h3)≥L(y, x−1, `2)∧L(u, `2, h3)> θ.

(55)

From (54) and (55), because of Deﬁnition 1 (ii),

we conclude that h3=vwhich implies that

L(u, `2, v)> θ. Consequently,

(u∗`1)◦(u∗`2)(e)≥

L(u, `1, t)∧L(u, `2, v)∧T(t, v, e)> θ.

(56)

Let k∈Xsuch that L(u, `, k)> θ, it follows

that

u∗(`1◦`2)(k)≥L(`1, `1, `)∧L(u, `, k)> θ.

(57)

According to Deﬁnition 1 (iii), (56) and (57) as-

sure k=eand then (u∗`)(e) = L(u, `, e)> θ.

Once again Lemma 5 shows that `=e, and hence

x∈ZF(X). This proves the desired result.

Remark 2. If we consider that (X, T, L)is a

fuzzy right near-ring, we can follow the same ar-

guments as those used previously, taking into ac-

count the second condition in Lemmas 2 and 3.

As an application of Theorems 2 and 3, we get

the following result.

Theorem 4. Let (X, T, L)be a fuzzy near-ring.

If ZF(X)contains a nontrivial fuzzy semigroup

ideal I, then (X, T, L)is a fuzzy commutative

ring.

Proof. We divide the proof into four essential

steps.

•In the ﬁrst part, we assume that (X, T, L) is a

fuzzy left near-ring, and we prove that property

(iv) of Deﬁnition 1 is satisﬁed.

Firstly, showing that ∀x, y, t1, t2∈Xand z∈I,

if (x◦y)∗z(t1)> θ and (x∗z)◦(y∗z)(t2)> θ,

then t1=t2.

In fact, let x, y, t1, t2∈Xand z∈Isatisfy

(x◦y)∗z(t1)> θ and (x∗z)◦(y∗z)(t2)> θ.

Choosing h1∈Xsuch that T(x, y, h1)> θ and

using the fact that (x◦y)∗z(t1)> θ, we obtain

L(h1, z, t1)> θ and in view of z∈I⊆ZF(X),

we ﬁnd that L(z, h1, t1)> θ.

Similarly, let h2, h3∈Xsuch that L(x, z, h2)> θ

and L(y, z, h3)> θ and since (x∗z)◦(y∗z)(t2)>

θ, we conclude that T(h2, h3, t2)> θ.

Also, as z∈I⊆ZF(X), we have L(z, x, h2)> θ

and L(z, y, h3)> θ. Then,

(z∗x)◦(z∗y)(t2)≥

L(z, x, h2)∧L(z, y, h3)∧T(h2, h3, t2)> θ,

(58)

and

z∗(x◦y)(t1)≥T(x, y, h1)∧L(z, h1, t1)> θ.

(59)

Invoking Deﬁnition 1 (iii), the last two relations

give t1=t2.

•Secondly, checking that ∀x, y, t, t1, t2∈Xif

(x◦y)∗t(t1)> θ and (x∗t)◦(y∗t)(t2)> θ,

then t1=t2.

For this purpose, let x, y, t, t1, t2∈Xsuch that

(x◦y)∗t(t1)> θ and (x∗t)◦(y∗t)(t2)> θ.

Let z∈Iand taking s∈Xsuch that (t∗z)(s)>

θ, by deﬁning I, we get s∈I.

Let `, h1, h2, h, v ∈Xsuch that T(x, y, `)> θ,

L(x, s, h1)> θ,L(y, s, h2)> θ,L(`, s, h)> θ and

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T(h1, h2, v)> θ. Then,

(x◦y)∗s(h)≥T(x, y, `)∧L(`, s, h)> θ, (60)

and

(x∗s)◦(y∗s)(v)≥

L(x, s, h1)∧L(y, s, h2)∧T(h1, h2, v)> θ.

(61)

The previous step guarantees h=v, so that

T(h1, h2, h)> θ.

Also, we have

(`∗(t∗z))(h)≥L(t, z, s)∧L(`, s, h)> θ, (62)

and from T(x, y, `)> θ together (x◦y)∗t(t1)>

θ, we get L(`, t, t1)> θ.

Now, taking k∈Xsuch that L(t1, z, k)> θ. It

follows that,

(`∗t)∗z(k)≥L(`, t, t1)∧L(t1, z, k)> θ. (63)

Applying Deﬁnition 1 (ii), for (62) and (63), we

get k=hand hence,

L(t1, z, h)> θ. (64)

On the other hand, we have

x∗(t∗z)(h1)≥L(t, z, s)∧L(x, s, h1)> θ (65)

and

y∗(t∗z)(h2)≥L(t, z, s)∧L(y, s, h2)> θ. (66)

Considering `1, `2, m, n ∈Xsatisfy L(x, t, `1)>

θ,L(y, t, `2)> θ,L(`1, z, m)> θ and L(`2, z, n)>

θ. Then, we can see that (x∗t)∗z(m)> θ

and (y∗t)∗z(n)> θ. Combining the last two

results with (65) and (66), respectively, and apply

Deﬁnition 1 (ii), we arrive at m=h1and n=h2

which give L(`1, z, h1)> θ and L(`2, z, h2)> θ.

Accordingly,

(`1∗z)◦(`2∗z)(h)≥

L(`1, z, h1)∧L(`2, z, h2)∧T(h1, h2, h)> θ.

(67)

As well, from L(x, t, `1)> θ,L(y, t, `2)> θ and

(x∗t)◦(y∗t)(t2)> θ, we get T(`1, `2, t2)> θ.

Let h0∈Xsuch that L(t2, z, h0)> θ, then

(`1◦`2)∗z)(h0)≥L(t2, z, h0)∧T(`1, `2, t2)> θ.

(68)

Since z∈I, then in view of the ﬁrst part, (67)

and (68) assure that h=h0which implies that

L(t2, z, h)> θ. (69)

Let t∈Xsuch that L(e, z, t)> θ, then

(e◦e)∗z(t)≥T(e, e, e)∧L(e, z, t)> θ. (70)

Consider v∈Xsuch that T(t, t, v)> θ, we have

(e∗z)◦(e∗z)(v)≥

L(e, z, t)∧L(e, z, t)∧T(t, t, v)> θ.

(71)

Using the conclusion of the ﬁrst part, we obtain

v=twhich means that T(t, t, t)> θ and hence

t=eby Lemma 1 (4). Thus, L(e, z, e)> θ.

Now, let t3∈Xsuch that L(`, t−1, t3)> θ which

means (x◦y)∗t−1(t3)> θ. From lemma 4 (i),

we have `∗t◦`∗t−1(e)> θ. Using our

hypotheses that L(`, t, t1)> θ and L(`, t−1, t3)>

θ, we ﬁnd that

(t1◦t3)(e) = T(t1, t3, e)> θ, (72)

that is,

(t1◦t3)∗z(e)≥T(t1, t3, e)∧L(e, z, e)> θ. (73)

Let s1, p ∈Xsuch that L(t3, z, s1)> θ and

T(h, s1, p)> θ and invoking (64), we obtain

(t1∗z)◦(t3∗z)(p)≥

L(t1, z, h)∧L(t3, z, s1)∧T(h, s1, p)> θ.

(74)

In view of the preceding step, we conclude that

p=e, and therefore T(h, s1, e)> θ.

By using (69), we get

(t2∗z)◦(t3∗z(e)≥

L(t2, z, h)∧L(t3, z, s1)∧T(h, s1, e)> θ,

(75)

Now, choosing v1, v2∈Xsuch that T(t2, t3, v1)>

θand L(v1, z, v2)> θ, then

(t2◦t3)∗z(v2)> θ. (76)

Once again from the previous step, we get v2=e,

so that (v1∗z)(e)> θ which, because of Lemma

5 (i), gives v1=e. And therefore,

(t2◦t3)(e) = T(t2, t3, e)> θ. (77)

Applyiny Lemma 1 (2) to (72) and (77), we ﬁnd

that t1=t2.

We can use the same arguments to show the

property (iii) of Deﬁnition 1 when (X, T, L) is a

fuzzy right near-ring.

•Thirdly, showing that (X, T ) is commuta-

tive. The fact that Iis nontrivial ideal assures

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the existence of x, y, t ∈I, with t6=e, such

that (x∗y)(t)> θ. Indeed, suppose that for all

x, y, t ∈I, where t6=e, we have (x∗y)(t)≤θ,

then inevitably we will have for all x, y ∈X,

(x∗y)(e)> θ, and thus Lemma 5 forces that x=e

which means that I={e}, leading to a contra-

diction with the fact that Iis a nontrivial ideal of

X. Consequently, there exist x, y, t ∈Isuch that

(x∗y)(t)> θ and t6=e. Since t∈I⊆ZF(X)

and t6=e, then t∈(ZF(X))∗.

Let v∈Xsuch that T(t, t, v)> θ. Our goal is to

show that v∈ZF(X). We have

(x∗y)◦(x∗y)(v)≥

L(x, y, t)∧L(x, y, t)∧T(t, t, v)> θ.

(78)

Now, letting h, k ∈Xsatisfy T(y, y, h)> θ and

L(x, h, k)> θ, then

x∗(y◦y)(k)≥T(y, y, h)∧L(x, h, k)> θ. (79)

Deﬁnition 1 (iii) assures that k=vand therefore

L(x, h, v)> θ. Taking into account that x∈

Iand h∈Xwe conclude that v∈I, so that

v∈ZF(X) and hence (X, T ) is commutative by

Theorem 2.

•Finally, to complete the proof of this theorem,

we show that (X, T, L) is commutative. For this,

let x∈Xand u∈I, then there exists t∈Xsuch

that L(x, u, t)> θ. In view of u∈I⊆ZF(X),

we obtain L(u, x, t)> θ. Then,

∀u∈I, ∃t∈Xsuch that L(x, u, t)> θ and

L(u, x, t)> θ.

By application of Theorem 3, we ﬁnd x∈ZF(X)

and hence, (X, T, L) is a commutative fuzzy ring.

4 Conclusion

In this paper, a new type of fuzzy semigroup ideal

was created and some of its related properties

were studied analogously to the ordinary semi-

group ideal. Also, using this new deﬁnition, we

proved that under some other conditions, a fuzzy

near ring must be a fuzzy commutative ring. The

practical applications of our study will be the sub-

ject of future research.

Acknowledgment:

The authors thank the reviewer for valuable

suggestions and comments.

References:

[1] J. N. Mordeson and D. S. Malik, Fuzzy

commutative algebra, World Scientiﬁc

Publishing Co. Pte. Ltd. 1998.

[2] L. A. Zadeh, Fuzzy sets, Inform. Control, 8

(1965), pp. 338-353.

[3] Esmaeel ranjbar-yanehsari and Mohsen

asghari-larimi, A new view of fuzzy vector

space over fuzzy ﬁeld, Jordan Journal of

Mathematics and Statistics, 11(3)

(2018),193 - 210.

[4] M. A. ztrk, Y. B. Jun and H. Yazarh, A new

view of fuzzy gamma rings, Hacet. J. Math.

Stat., 39 (3) (2010), pp. 365-378.

[5] A. Rosenfeld, Fuzzy groups, J. Math. Anal.

Appl., 35 (1971), pp. 512-517.

[6] X. Yuan, E. S. Lee, Fuzzy group based on

fuzzy binary operation, Comput. Math.

App., 47 (2004), pp. 631-641.

[7] H. Akta and N. agman, A type of fuzzy ring,

Arch. Math. Logic, 46 (2007), pp. 165-177.

[8] G. Pilz, Near-Rings. 2nd Edition, North

Holland /American Elsevier, Amsterdam,

1983.

[9] M. Oukessou, A. Raji and M. Ou-mha, study

of the fuzzy multiplicative center in a special

class of fuzzy near rings, Boletim da

Sociedade Paranaense de Matemtica, in

press.

[10] H. E. Bell, On derivations in near-rings II,

Kluwer Academic Publishers Netherlands,

(1997), pp. 191-197.

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self:

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Conﬂict of Interest:

There is no conﬂict of interest.

Contribution of individual authors:

The authors contributed equally to this work.

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