Open Access

Fuzzy prime ideals redefined

Journal of Inequalities and Applications20122012:203

https://doi.org/10.1186/1029-242X-2012-203

Received: 24 April 2012

Accepted: 3 September 2012

Published: 19 September 2012

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.

Keywords

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

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 = { x G A ( 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 x X , 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 q A ), if
A ( x ) t ( resp.  A ( x ) + t > 1 ) ,

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

In the following discussions, R always stands for an associate ring, λ and μ are constant numbers such that 0 λ < μ 1 , 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 , y R and t , r ( 0 , 1 ] ,
  1. (1)

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

     
  2. (2)

    x t A ( x ) t q A ,

     
  3. (3)

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

     

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 , y R ( x y ) t , ( y x ) t q A , 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 , y R ,
  1. (1)

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

     
  2. (2)

    A ( x y ) A ( x ) A ( y ) 0.5 ((2) A ( x y ) ( 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 , y R ,
A ( x + y ) λ A ( x ) A ( y ) μ , A ( x ) λ A ( 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 , y R , A ( x y ) λ A ( 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 , y R ,
  1. (1)

    A ( x y ) λ A ( x ) A ( y ) μ ,

     
  2. (2)

    A ( x y ) λ A ( 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 , y R ,
  1. (1)

    A ( x y ) λ A ( x ) A ( y ) μ ,

     
  2. (2)

    A ( x y ) λ A ( y ) μ (resp. A ( x y ) λ A ( 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 , y R ,
  1. (1)

    A ( x y ) λ A ( x ) A ( y ) μ ,

     
  2. (2)

    A ( x y ) 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 A B 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 } , x R .

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

For all x , y R , 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 A B is defined as follows: x R ,
( A B ) ( 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 A B is a ( λ , μ ) -fuzzy left ideal of R.

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

So, A B 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 A B 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 A B 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 ( x y ) λ A ( x ) A ( y ) μ > t . Considering λ t , we have A ( x y ) > t and x y A t . Similarly, x y 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 ( x y ) λ A ( x ) A ( y ) μ holds for all x , y R .

Similarly, A ( x y ) λ A ( x ) A ( y ) μ , x , y R . 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 ) ) , x R .

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 q A 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 ] ,
A B 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 ) λ t μ and B ( x ) λ t μ , then ( A B ) ( x ) λ t μ . So, x ( A B ) t ( λ , μ ) .

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

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

Case 4, if A ( x ) > ( 2 μ t ) λ and B ( x ) λ t μ , 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 ) λ t μ or A ( x ) B ( x ) > ( 2 μ t ) λ . It means that A ( x ) λ t μ , or B ( x ) λ t μ , or A ( x ) > ( 2 μ t ) λ , or B ( x ) > ( 2 μ t ) 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 t λ , then A t ( λ , μ ) = R . If λ < t μ , 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 ( 2 μ t ) 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 a b S a S or b S .

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 x R and t ( 0 , 1 ] , ( x 2 ) t A x t q A ,

     
  2. (2)

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

     
  3. (3)

    fuzzy semiprimary if for all x , y R and t ( 0 , 1 ] , ( x y ) t A x t m q A or y t n q A for some m , n N ,

     
  4. (4)

    fuzzy primary if for all x , y R and t ( 0 , 1 ] , ( x y ) t A x t q A or y t m q A for some m N .

     

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 x R and t ( 0 , 1 ] , ( x 2 ) t A x t ( λ , μ ) A ,

     
  2. (2)

    ( λ , μ ) -fuzzy prime if for all x , y R 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 , y R and t ( 0 , 1 ] , ( x y ) t A x t m ( λ , μ ) A or y t n ( λ , μ ) A for some m , n N ,

     
  4. (4)

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

     

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 , y R ,
λ A ( x ) A ( y ) A ( x y ) μ .

Proof Let A be ( λ , μ ) -fuzzy prime. If possible, let λ A ( 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, λ A ( x 0 ) < t = t μ and λ A ( y 0 ) < t = t μ . From λ A ( x 0 ) < t μ , we have A ( x 0 ) λ A ( x 0 ) < t μ 2 μ t and hence x 0 A t ( λ , μ ) . Similarly, y 0 A t ( λ , μ ) . This is a contradiction. Therefore, for all x , y R , we have λ A ( x ) A ( y ) A ( x y ) μ .

Conversely, if for all x , y R , λ A ( x ) A ( y ) A ( x y ) μ and ( x y ) t A , then λ A ( x ) A ( y ) t μ . So, λ A ( x ) t μ or λ A ( y ) t μ . 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 x R ,
λ A ( 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 , y R , m 0 N such that
λ A ( x ) A ( y m 0 ) A ( x y ) μ .

Proof Let A be ( λ , μ ) -fuzzy primary. If possible, there exist x 0 , y 0 R such that λ A ( x 0 ) A ( y 0 m ) < A ( x 0 y 0 ) μ for all m N , then λ A ( x 0 ) A ( y 0 m ) < t μ and ( x 0 y 0 ) t A , where t = A ( x 0 y 0 ) μ . So, λ A ( x 0 ) < t = t μ and λ A ( y 0 m ) < t = t μ . From λ A ( x 0 ) < t , we have A ( x 0 ) λ A ( x 0 ) < t μ 2 μ t and hence x 0 A t ( λ , μ ) . Similarly, y 0 m A t ( λ , μ ) . This is a contradiction. Therefore, for all x , y R , m 0 N such that λ A ( x ) A ( y m 0 ) A ( x y ) μ .

Conversely, if for all x , y R , m 0 N such that λ A ( x ) A ( y m 0 ) A ( x y ) μ , then from ( x y ) t A , we have λ A ( x ) A ( y m 0 ) t μ . So, λ A ( x ) t μ or λ A ( y m 0 ) t μ . 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 , y R , m 0 , n 0 N such that
λ A ( x m 0 ) A ( y n 0 ) A ( x y ) 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 x y A t , then λ A ( x ) A ( y ) A ( x y ) μ t μ = 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 t λ , 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 x y 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 = t μ x 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 A B 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 A B 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 A B , 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 A B 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 A B 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 , y R 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.

Declarations

Acknowledgements

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

Authors’ Affiliations

(1)
School of Mathematics Science, Liaocheng University

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