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Fuzzy prime ideals redefined

Abstract

In order to generalize the notions of a (,q)-fuzzy subring and various (,q)-fuzzy ideals of a ring, a (λ,μ)-fuzzy subring and a (λ,μ)-fuzzy ideal of a ring are defined. The concepts of (λ,μ)-fuzzy semiprime, prime, semiprimary and primary ideals are introduced, and the characterizations of such fuzzy ideals are obtained based on a (λ,μ)-cut set.

MSC:16L99, 03E72.

1 Introduction

The concept of a fuzzy set introduced by Zadeh [1] was applied to the group theory by Rosenfeld [2] and the ring theory by Liu [3]. Since then, many scholars have studied the theories of fuzzy subrings and various fuzzy prime ideals [46]. It is worth pointing out that Bhakat and Das introduced the concept of an (α,β)-fuzzy subgroup by using the ‘belongs to’ relation and ‘quasi-coincident with’ relation between a fuzzy point and a fuzzy subset, and gave the concepts of an (,q)-fuzzy subgroup and an (,q)-fuzzy subring [7, 8]. It is well known that a fuzzy subset A of a group G is a Rosenfeld’s fuzzy subgroup if and only if A t ={xGA(x)t} is a subgroup of G for all t(0,1] (for our convenience, here is regarded as a subgroup of G). Similarly, A is an (,q)-fuzzy subgroup if and only if A t is a subgroup of G for all t(0,0.5]. A corresponding result should be considered when A t is a subgroup of G for all t(a,b], where (a,b] is an arbitrary subinterval of [0,1]. Motivated by the above problem, Yuan et al. [9] introduced a fuzzy subgroup with the thresholds of a group. In order to generalize the concepts of an (,q)-fuzzy subring and an (,q)-fuzzy ideal of a ring, Yao [10] introduced the notions of a (λ,μ)-fuzzy subring and a (λ,μ)-fuzzy ideal and discussed their fundamental properties. In this paper, we will introduce the concepts of (λ,μ)-fuzzy prime, fuzzy semiprime, fuzzy primary and fuzzy semiprimary ideals of a ring.

2 (λ,μ)-fuzzy ideal

Let X be a nonempty set. By a fuzzy subset A of X, we mean a map from X to the interval [0,1], A:X[0,1]. If A is a fuzzy subset of X and t[0,1], then the cut set A t and the open cut set A t of A are defined as follows:

A t = { x X A ( x ) t } , A t = { x X A ( x ) > t } .

First, we recall some definitions and results for the sake of completeness.

Definition 1 Let xX, t(0,1]. A fuzzy subset A of X of the form

A(y)={ t , if  y = x , 0 , if  y x

is said to be a fuzzy point with support x and value t and is denoted by x t .

Definition 2 [11]

A fuzzy point x t is said to belong to (resp. be quasi-coincident with) a fuzzy subset A, written as x t A (resp. x t qA), if

A(x)t ( resp.  A ( x ) + t > 1 ) ,

x t A or x t qA will be denoted by x t qA.

In the following discussions, R always stands for an associate ring, λ and μ are constant numbers such that 0λ<μ1, and N denotes the set of all positive integers.

Definition 3 [8]

A fuzzy subset A of R is said to be an (,q)-fuzzy subring of R if for all x,yR and t,r(0,1],

  1. (1)

    x t , y r A ( x + y ) t r qA,

  2. (2)

    x t A ( x ) t qA,

  3. (3)

    x t , y r A ( x y ) t r qA.

Definition 4 [8]

A fuzzy subset A of R is said to be an (,q)-fuzzy ideal of R if

  1. (1)

    A is an (,q)-fuzzy subring of R,

  2. (2)

    x t A,yR ( x y ) t , ( y x ) t qA,t(0,1].

According to Definition 3 and Definition 4, we have that a fuzzy subset A of R is a (λ,μ)-fuzzy subring (ideal) of R if and only if for all x,yR,

  1. (1)

    A(xy)A(x)A(y)0.5,

  2. (2)

    A(xy)A(x)A(y)0.5 ((2) A(xy)(A(x)A(y))0.5).

In order to give more general concepts of a fuzzy subring and a fuzzy ideal of R, we introduce the following definitions.

Definition 5 A fuzzy subset A of R is said to be a (λ,μ)-fuzzy addition subgroup of R if for all x,yR,

A(x+y)λA(x)A(y)μ,A(x)λA(x)μ.

Clearly, a fuzzy subset A of R is a (λ,μ)-fuzzy addition subgroup of R if and only if for all x,yR, A(xy)λA(x)A(y)μ.

Definition 6 [10]

A fuzzy subset A of R is said to be a (λ,μ)-fuzzy subring of R if for all x,yR,

  1. (1)

    A(xy)λA(x)A(y)μ,

  2. (2)

    A(xy)λA(x)A(y)μ.

Definition 7 [10]

A fuzzy subset A of R is said to be a (λ,μ)-fuzzy left ideal (resp. fuzzy right ideal) of R if for all x,yR,

  1. (1)

    A(xy)λA(x)A(y)μ,

  2. (2)

    A(xy)λA(y)μ (resp. A(xy)λA(x)μ).

A is said to be a (λ,μ)-fuzzy ideal of R if it is both a (λ,μ)-fuzzy left ideal and a (λ,μ)-fuzzy right ideal of R.

According to the above definitions, a (λ,μ)-fuzzy left ideal or a (λ,μ)-fuzzy right ideal of R must be a (λ,μ)-fuzzy subring. A fuzzy subset A of R is a (λ,μ)-fuzzy ideal of R if and only if for all x,yR,

  1. (1)

    A(xy)λA(x)A(y)μ,

  2. (2)

    A(xy)λ(A(x)A(y))μ.

Obviously, an (,q)-fuzzy subring (fuzzy ideal) of R is a (λ,μ)-fuzzy subring (fuzzy ideal) of R with λ=0 and μ=0.5.

The following theorem is obvious.

Theorem 1 Let A, B be (λ,μ)-fuzzy left ideals (fuzzy right ideals, fuzzy ideals, fuzzy subrings) of R. Then AB is also a fuzzy left ideal (fuzzy right ideal, fuzzy ideal, fuzzy subring) of R.

Theorem 2 Let A, B be (λ,μ)-fuzzy left ideals (fuzzy right ideals, fuzzy ideals) of R. Then A+B is also a (λ,μ)-fuzzy left ideal (fuzzy right ideal, fuzzy ideal) of R, where

(A+B)(x)=sup { A ( x 1 ) B ( x 2 ) x = x 1 + x 2 } ,xR.

Proof We only prove the case of a (λ,μ)-fuzzy left ideal.

For all x,yR, we have

So, A+B is a (λ,μ)-fuzzy left ideal of R.

Let A, B be fuzzy subsets of R. Then the fuzzy subset AB is defined as follows: xR,

(AB)(x)={ sup { inf 1 i n ( A ( x i ) B ( y i ) ) x = i = 1 n x i y i , x i , y i R , n N } , 0 , if  x  can be expressed as  x = x i y i , x i , y i R , 0 , otherwise .

 □

Theorem 3 Let A be a (λ,μ)-fuzzy left ideal, and let B be a fuzzy subset of R. Then AB is a (λ,μ)-fuzzy left ideal of R.

Proof For all z 1 , z 2 R, we have

So, AB is a (,q)-fuzzy left ideal of R. □

Similarly, we have the following theorem.

Theorem 4 Let A be a fuzzy subset, and let B be a (λ,μ)-fuzzy right ideal of R. Then AB is a (λ,μ)-fuzzy right ideal of R.

The following theorem is an immediate consequence of Theorem 3 and Theorem 4.

Theorem 5 Let A be a (λ,μ)-fuzzy left ideal, and let B be a (λ,μ)-fuzzy right ideal of R. Then AB is a (λ,μ)-fuzzy ideal of R.

One of the most common methods of studying a fuzzy subring and a fuzzy ideal is by using their cut sets. Now we give the relation between a (λ,μ)-fuzzy subring (fuzzy ideal) with its cut set or open cut set.

Theorem 6 [10]

A fuzzy subset A of R is a (λ,μ)-fuzzy subring (fuzzy ideal) of R if and only if for all t(λ,μ], A t is a subring (ideal) of R or A t =.

Theorem 7 A fuzzy subset A of R is a (λ,μ)-fuzzy subring (fuzzy ideal) of R if and only if for all t[λ,μ), A t is a subring (ideal) of R or A t =.

Proof We only prove the case of a (λ,μ)-fuzzy subring.

Let A be a (λ,μ)-fuzzy subring of R, and let t[λ,μ). If x,y A t , then A(x)A(y)>t, A(xy)λA(x)A(y)μ>t. Considering λt, we have A(xy)>t and xy A t . Similarly, xy A t . It follows that A t is a subring of R.

Conversely, assume that A t is a subring of R for all t[λ,μ). If possible, let A( x 0 y 0 )λ<A( x 0 )A( y 0 )μ for some x 0 , y 0 R. Put t=A( x 0 y 0 )λ, then t[λ,μ), and A( x 0 y 0 )t, A( x 0 )A( y 0 )>t. So, x 0 , y 0 A t , and x 0 y 0 A t . This is a contradiction to the fact that A t is a subring of R. This shows that A(xy)λA(x)A(y)μ holds for all x,yR.

Similarly, A(xy)λA(x)A(y)μ, x,yR. That is, A t is a subring of R. □

In the following theorems, it is shown that the homomorphism image (preimage) of a (λ,μ)-fuzzy subring is also a (λ,μ)-fuzzy subring. Similar result can be obtained for a (λ,μ)-fuzzy ideal under some conditions.

Theorem 8 [10]

Let f:R R be a homomorphism of rings. If A is a (λ,μ)-fuzzy subring of R, then f(A) is a (λ,μ)-fuzzy subring of R . Particularly, if A is a (λ,μ)-fuzzy ideal of R and f is onto, then f(A) is a (λ,μ)-fuzzy ideal of R , where

f(A)(y)={ sup { A ( x ) f ( x ) = y } , if f 1 ( y ) , 0 , otherwise y R .

Theorem 9 [10]

Let f:R R be a homomorphism of rings. If B is a (λ,μ)-fuzzy subring (fuzzy ideal) of R , then f 1 (B) is a (λ,μ)-fuzzy subring (fuzzy ideal) of R, where

f 1 (B)(x)=B ( f ( x ) ) ,xR.

3 (λ,μ)-cut set

Based on the notion of an (,q)-level subset defined in [12], we introduce the concept of a (λ,μ)-cut set of a fuzzy subset. Let A be a fuzzy subset of a set X and t[0,1]. Then the subset A t ( λ , μ ) of X defined by

A t ( λ , μ ) = { x X A ( x ) λ t μ  or  A ( x ) > ( 2 μ t ) λ }

is said to be a (λ,μ)-cut set of A. We denote x t ( λ , μ ) A if x A t ( λ , μ ) .

Obviously, if A is a fuzzy subset of X and t[0,1], then

A t ( λ , μ ) ={ X , t λ , A t , λ < t μ , A ( 2 μ t ) λ , t > μ .

Moreover, x t qA coincides with x t ( 0 , 0.5 ) A, and x t q k A [12] coincides with x t ( 0 , k 2 ) A.

Lemma 1 Let A, B be fuzzy subsets of X. Then for all t[0,1],

AB A t ( λ , μ ) B t ( λ , μ ) .

Proof The proof is straightforward. □

Theorem 10 Let A, B be fuzzy subsets of X and t[0,1]. Then

  1. (1)

    ( A B ) t ( λ , μ ) = A t ( λ , μ ) B t ( λ , μ ) ,

  2. (2)

    ( A B ) t ( λ , μ ) = A t ( λ , μ ) B t ( λ , μ ) .

Proof

(1) Obviously, we have ( A B ) t ( λ , μ ) A t ( λ , μ ) B t ( λ , μ ) from Lemma 1. If x A t ( λ , μ ) B t ( λ , μ ) , then x A t ( λ , μ ) and x B t ( λ , μ ) . So, we have the following four cases.

Case 1, if A(x)λtμ and B(x)λtμ, then (AB)(x)λtμ. So, x ( A B ) t ( λ , μ ) .

Case 2, if A(x)>(2μt)λ and B(x)>(2μt)λ, then (AB)(x)>(2μt)λ. So, x ( A B ) t ( λ , μ ) .

Case 3, if A(x)λtμ and B(x)>(2μt)λ, then when tμ, we have B(x)λ>(2μt)λμtμ and hence (AB)(x)λtμ. When t>μ, we have A(x)μ>(2μt)λ and hence (AB)(x)>(2μt)λ. It means that x ( A B ) t ( λ , μ ) .

Case 4, if A(x)>(2μt)λ and B(x)λtμ, then we can also obtain that x ( A B ) t ( λ , μ ) just as in Case 3.

Therefore, ( A B ) t ( λ , μ ) = A t ( λ , μ ) B t ( λ , μ ) .

(2) Obviously, we have ( A B ) t ( λ , μ ) A t ( λ , μ ) B t ( λ , μ ) from Lemma 1.

Let x ( A B ) t ( λ , μ ) , then either A(x)B(x)λtμ or A(x)B(x)>(2μt)λ. It means that A(x)λtμ, or B(x)λtμ, or A(x)>(2μt)λ, or B(x)>(2μt)λ. So x A t ( λ , μ ) or x B t ( λ , μ ) . That is, x A t ( λ , μ ) B t ( λ , μ ) . Hence, ( A B ) t ( λ , μ ) = A t ( λ , μ ) B t ( λ , μ ) . □

Theorem 11 Let A, B, C be fuzzy subsets of X and t[0,1]. Then

  1. (1)

    [ A ( B C ) ] t ( λ , μ ) = ( A B ) t ( λ , μ ) ( A C ) t ( λ , μ ) ,

  2. (2)

    [ A ( B C ) ] t ( λ , μ ) = ( A B ) t ( λ , μ ) ( A C ) t ( λ , μ ) .

Proof The proof can be obtained immediately from Theorem 10. □

Theorem 12 Let A be a (λ,μ)-fuzzy subring (fuzzy ideal) of R. Then for all t[0,1], A t ( λ , μ ) is a subring (ideal) of R or A t ( λ , μ ) =.

Proof We only prove the case of a (λ,μ)-fuzzy subring.

If tλ, then A t ( λ , μ ) =R. If λ<tμ, then A t ( λ , μ ) = A t and A t is a subring of R from Theorem 6. If t>μ, then A t ( λ , μ ) = A ( 2 μ t ) λ and (2μt)λ[λ,μ). So, A t ( λ , μ ) is a subring of R from Theorem 7. □

Theorem 13 Let A be a fuzzy subset of R. If for all t(λ,μ], A t ( λ , μ ) is a subring (ideal) of R or A t ( λ , μ ) =, then A is a (λ,μ)-fuzzy subring (fuzzy ideal) of R.

Proof The proof can be obtained from Theorem 6. □

4 (λ,μ)-fuzzy prime and fuzzy primary ideal

There are several deferent definitions of a fuzzy prime ideal and a fuzzy primary ideal of R. In this section, by a prime ideal S of R, we mean an ideal of R such that abSaS or bS.

Bhakat and Das [8] defined fuzzy prime, fuzzy semiprime, fuzzy primary and fuzzy semiprimary ideals in a ring which must be (,q)-fuzzy ideals first.

Definition 8 [8]

An (,q)-fuzzy ideal A of R is said to be

  1. (1)

    fuzzy semiprime if for all xR and t(0,1], ( x 2 ) t A x t qA,

  2. (2)

    fuzzy prime if for all x,yR and t(0,1], ( x y ) t A x t qA or y t qA,

  3. (3)

    fuzzy semiprimary if for all x,yR and t(0,1], ( x y ) t A x t m qA or y t n qA for some m,nN,

  4. (4)

    fuzzy primary if for all x,yR and t(0,1], ( x y ) t A x t qA or y t m qA for some mN.

In order to generalize these notions, we introduce (λ,μ)-fuzzy prime, (λ,μ)-fuzzy semiprime, (λ,μ)-fuzzy primary and (λ,μ)-fuzzy semiprimary ideals.

Definition 9 A (λ,μ)-fuzzy ideal A of R is said to be

  1. (1)

    (λ,μ)-fuzzy semiprime if for all xR and t(0,1], ( x 2 ) t A x t ( λ , μ ) A,

  2. (2)

    (λ,μ)-fuzzy prime if for all x,yR and t(0,1], ( x y ) t A x t ( λ , μ ) A or y t ( λ , μ ) A,

  3. (3)

    (λ,μ)-fuzzy semiprimary if for all x,yR and t(0,1], ( x y ) t A x t m ( λ , μ ) A or y t n ( λ , μ ) A for some m,nN,

  4. (4)

    (λ,μ)-fuzzy primary if for all x,yR and t(0,1], ( x y ) t A x t ( λ , μ ) A or y t m ( λ , μ ) A for some mN.

In the following four theorems, we give the equivalence condition of a (λ,μ)-fuzzy prime (semiprime, primary, semiprimary) ideal.

Theorem 14 A (λ,μ)-fuzzy ideal A of R is (λ,μ)-fuzzy prime if and only if for all x,yR,

λA(x)A(y)A(xy)μ.

Proof Let A be (λ,μ)-fuzzy prime. If possible, let λA( x 0 )A( y 0 )<A( x 0 y 0 )μ for some x 0 , y 0 R. Put t=A( x 0 y 0 )μ, then A( x 0 )A( y 0 )<t and ( x 0 y 0 ) t A. So, λA( x 0 )<t=tμ and λA( y 0 )<t=tμ. From λA( x 0 )<tμ, we have A( x 0 )λA( x 0 )<tμ2μt and hence x 0 A t ( λ , μ ) . Similarly, y 0 A t ( λ , μ ) . This is a contradiction. Therefore, for all x,yR, we have λA(x)A(y)A(xy)μ.

Conversely, if for all x,yR, λA(x)A(y)A(xy)μ and ( x y ) t A, then λA(x)A(y)tμ. So, λA(x)tμ or λA(y)tμ. That is, x ( λ , μ ) A or y ( λ , μ ) A. Hence, A is (λ,μ)-fuzzy prime. □

Theorem 15 A (λ,μ)-fuzzy ideal A of R is (λ,μ)-fuzzy semiprime if and only if for all xR,

λA(x)A ( x 2 ) μ.

Proof The proof is similar to that of Theorem 14. □

Theorem 16 A (λ,μ)-fuzzy ideal A of R is (λ,μ)-fuzzy primary if and only if for all x,yR, m 0 N such that

λA(x)A ( y m 0 ) A(xy)μ.

Proof Let A be (λ,μ)-fuzzy primary. If possible, there exist x 0 , y 0 R such that λA( x 0 )A( y 0 m )<A( x 0 y 0 )μ for all mN, then λA( x 0 )A( y 0 m )<tμ and ( x 0 y 0 ) t A, where t=A( x 0 y 0 )μ. So, λA( x 0 )<t=tμ and λA( y 0 m )<t=tμ. From λA( x 0 )<t, we have A( x 0 )λA( x 0 )<tμ2μt and hence x 0 A t ( λ , μ ) . Similarly, y 0 m A t ( λ , μ ) . This is a contradiction. Therefore, for all x,yR, m 0 N such that λA(x)A( y m 0 )A(xy)μ.

Conversely, if for all x,yR, m 0 N such that λA(x)A( y m 0 )A(xy)μ, then from ( x y ) t A, we have λA(x)A( y m 0 )tμ. So, λA(x)tμ or λA( y m 0 )tμ. That is, x t ( λ , μ ) A or y t m 0 ( λ , μ ) A. Hence, A is (λ,μ)-fuzzy primary. □

Theorem 17 A (λ,μ)-fuzzy ideal A of R is (λ,μ)-fuzzy semiprimary if and only if for all x,yR, m 0 , n 0 N such that

λA ( x m 0 ) A ( y n 0 ) A(xy)μ.

Proof The proof is similar to that of Theorem 16. □

Now, we characterize the (λ,μ)-fuzzy prime (semiprimary) ideal by using its cut set.

Theorem 18 A fuzzy subset A of R is a (λ,μ)-fuzzy prime (fuzzy semiprime) ideal if and only if for all t(λ,μ], A t is a prime (semiprime) ideal of R or A t =.

Proof We only prove the case of a (λ,μ)-fuzzy prime ideal.

Let A be a (λ,μ)-fuzzy prime ideal of R. Then A is a (λ,μ)-fuzzy ideal of R. So, A t is an ideal of R or A t = from Theorem 6. For all t(λ,μ], if xy A t , then λA(x)A(y)A(xy)μtμ=t. Considering λ<t, we have A(x)A(y)t. It follows that x A t or y A t . Hence, A t is a prime ideal of R.

Conversely, assume A t is a prime ideal of R for all t(λ,μ] whenever A t , then A t is an ideal of R, and hence A is a (λ,μ)-fuzzy ideal from Theorem 6. Let t(0,1] and ( x y ) t A. If tλ, then it is clear that x t ( λ , μ ) A. If t(λ,μ], then x A t or y A t since A t is a prime ideal of R. So, x t ( λ , μ ) A or y t ( λ , μ ) A. If t>μ, then xy A μ . It implies that x A μ or y A μ , since A μ is a prime ideal of R. Furthermore, we have

x A μ A(x)μ=tμx A t ( λ , μ ) x t ( λ , μ ) A.

Similarly,

y A μ y t ( λ , μ ) A.

It follows that A is a (λ,μ)-fuzzy prime ideal of R. □

Theorem 19 Let A be a (λ,μ)-fuzzy ideal of R such that A μ , and let B be a (λ,μ)-fuzzy prime ideal of A μ . Then AB is a (λ,μ)-fuzzy prime ideal of A μ .

Proof From Theorem 1 and Theorem 6, A μ is a subring of R and AB is a (λ,μ)-fuzzy ideal of A μ . For all x,y A μ , t(0,1], we have A(x)μ and A(y)μ. Hence, x,y A t ( λ , μ ) . If ( x y ) t AB, then x B t ( λ , μ ) or y B t ( λ , μ ) , since B is a (λ,μ)-fuzzy prime ideal of A μ . So x A t ( λ , μ ) B t ( λ , μ ) = ( A B ) t ( λ , μ ) , or y A t ( λ , μ ) B t ( λ , μ ) = ( A B ) t ( λ , μ ) . It follows that AB is a (λ,μ)-fuzzy prime ideal of A μ . □

Similarly, we have the following theorem.

Theorem 20 Let A be a (λ,μ)-fuzzy ideal of R such that A μ , and let B be a (λ,μ)-fuzzy semiprime (fuzzy primary, fuzzy semiprimary) ideal of A μ . Then AB is a (λ,μ)-fuzzy semiprime (fuzzy primary, fuzzy semiprimary) ideal of A μ .

The following theorem gives the relation between a (λ,μ)-fuzzy prime ideal with its preimage under a ring homomorphism.

Lemma 2 Let f:R R be a homomorphism of rings, and let B be a (λ,μ)-fuzzy subring of R . Then for all t(0,1], f 1 ( B ) t ( λ , μ ) = f 1 ( B t ( λ , μ ) ).

Theorem 21 Let f:R R be a homomorphism of rings, and let B be a (λ,μ)-fuzzy prime ideal of R . Then f 1 (B) is a (λ,μ)-fuzzy prime ideal of R.

Proof From Theorem 9, f 1 (B) is a (λ,μ)-fuzzy ideal of R. Let x,yR and t(0,1]. If ( x y ) t f 1 (B), then ( f ( x ) f ( y ) ) t B. Considering B is a (λ,μ)-fuzzy prime ideal of R , we have f(x) B t ( λ , μ ) or f(y) B t ( λ , μ ) . Hence x f 1 ( B ) t ( λ , μ ) or y f 1 ( B ) t ( λ , μ ) from Lemma 2. It follows that f 1 (B) is a (λ,μ)-fuzzy prime ideal of R. □

Similarly, we can obtain corresponding conclusions about a (λ,μ)-fuzzy semiprime ideal, a (λ,μ)-fuzzy primary ideal and a (λ,μ)-fuzzy semiprimary ideal. But in general, the homomorphism image f(A) of a (λ,μ)-fuzzy prime ideal A of R may not be (λ,μ)-fuzzy prime even if f is a surjective homomorphism.

5 Conclusion

In this paper, we proposed the concept of a (λ,μ)-fuzzy ideal which can be regarded as the generalization of a common fuzzy ideal introduced by Liu [11]. In the meantime, we also proposed several concepts of various (λ,μ)-fuzzy ideals such as a (λ,μ)-fuzzy prime ideal and a (λ,μ)-fuzzy primary ideal, and then we characterized their properties and obtained several equivalence conditions of a (λ,μ)-fuzzy prime ideal and a (λ,μ)-fuzzy primary ideal.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. 71140008).

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Yao, B. Fuzzy prime ideals redefined. J Inequal Appl 2012, 203 (2012). https://doi.org/10.1186/1029-242X-2012-203

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Keywords

  • (λ,μ)-fuzzy subring
  • (λ,μ)-fuzzy ideal
  • (λ,μ)-fuzzy prime ideal
  • (λ,μ)-fuzzy primary ideal