On the Fuzzy Nil Radicals of Fuzzy Ideals of κ-semirings

On the Fuzzy Nil Radicals of Fuzzy Ideals of k-semirings

Chang Bum Kim

Department of Mathematics, Kookmin University, Seoul 136-702, South Korea.

Abstract

We define and study the fuzzy nil radical of a fuzzy ideal of a commutative k−semiring.

Key Words : fuzzy nil radical, fuzzy prime radical, k−semiring, k−ideal, k−fuzzy ideal.

1. Introduction

Chun, Kim and Kim [2] constructed an extension of a k−semiring and studied a k−ideal of a k−semiring. The author et al. [3] constructed the quotient semiring of a k−semiring by a k−ideal. Liu [15] introduced and stud- ied the notion of fuzzy ideals of a ring. Following Liu, Mukherjee and Sen [18] defined and examined fuzzy prime ideals of a ring. Kumbhojkar and Bapat [6, 8] defined and studied the fuzzy nil radicals of a fuzzy ideal of a ring.

Yue [21] introduced the concept of a primary L−fuzzy ideal and a prime L− fuzzy ideal, and proved some fun- damental propositions. Primary fuzzy ideals were further investigated by Malik and Mordeson [17].

Kumar [9, 10, 11, 12, 13] extended the concept of a fuzzy ideal to a fuzzy semiprimary (semiprime, primary, prime, maximal) ideal in a ring. Also Malik and Morde- son [16] gave the necessary and sufficient conditions for a fuzzy subring or a fuzzy ideal A of a commutative ring R to be extended to one A ^e of a commutative ring S containing R as a subring.

Kim and Park [5] defined and studied the notion of the k−fuzzy ideal in a semiring, and they also introduced and studied the quotient semiring R/A of a k−semiring R by a k−fuzzy ideal A.

Kim [4] defined and studied the fuzzy maximal k−

ideals in a commutative k−semiring and characterized the quotient semiring R/A of a k−semiring R by a fuzzy max- imal(semiprimary, semiprime, primary, prime) k− ideal A, and he obtained some isomorphism theorems in a commu- tative k− semiring R.

In particular, Kumar [10], Malik and Mordeson [17], Sidky and Khatab [19] and Zadehi [23] defined and inves- tigated the fuzzy nil radicals of a fuzzy ideal in a ring.

The purpose of this paper is to define and study

the fuzzy nil radical of a fuzzy ideal of a commutative k−semiring with identity.

2. Preliminaries

In this section, we review some definitions and some results which will be used in the later section.

Definition 2-1(Chun, Kim and Kim [2]). A set R to- gether with associative binary operations called addition and multiplication (denoted by + and · respectively) will be called a semiring provided:

(1) addition is a commutative operation,

(2) there exists 0∈R such that x + 0 = x and x0 = 0x = 0 for each x∈R,

and

(3) multiplication distributes over addition both from the left and the right.

Definition 2-2(Chun, Kim and Kim [2]). A semiring R will be called a k−semiring if for any a, b∈R there ex- ists a unique element c in R such that either b = a + c or a = b + c but not both.

Definition 2-3(Chun, Kim and Kim [3]). A non-empty subset I of a semiring R is called a subsemiring if I is itself a semiring with respect to the binary operations defined in R. A subsemiring I is called an ideal of R if r∈R a∈I imply ar and ra∈I.

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Definition 2-4(Chun, Kim and Kim [3]). An ideal I of a semiring R is called a k−ideal if r + a∈I implies r∈I for each r∈R and each a∈I.

Let R be a k−semiring. Let R ^ be a set of the same car- dinality with R − {0} such that R∩R ^ = ∅ and let denote the image of a∈R − {0} under a given bijection by a ^ . Let

⊕ and  denote addition and multiplication respectively on a set ¯ R = R∪R ^ as follows:

a⊕b =

⎧ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎩

a + b if a, b∈R

(x + y) ^ if a = x ^ , b = y ^ ∈R ^

c if a∈R, b = y ^ ∈R ^ , a = y + c c ^ if a∈R, b = y ^ ∈R ^ , a + c = y, where c is the unique element in R such that either a = y + c or a + c = y but not both, and

ab =

⎧ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎩

ab if a, b∈R

xy if a = x ^ , b = y ^ ∈R ^ (ay) ^ if a∈R, b = y ^ ∈R ^ (xb) ^ if a = x ^ ∈R ^ , b∈R,

It can be shown that these operations are well defined.

Theorem 2-5(Chun, Kim and Kim [2]). If R is a k−semiring, then ( ¯ R, ⊕, ) is a ring, called an extension ring of R.

Remark . Let a denote the additive inverse of any element a∈R and write a⊕(b) simply as ab. Then it is clear that a ^ = a and a = a ^ for all a∈R.

Note that if R is a k−semiring with identity, then ¯ R is a ring with identity.

Theorem 2-6(Chun, Kim and Kim [2]). Let R be a k−semiring, I an ideal, and I ^ = {a ^ ∈R ^ |a∈I}. Then I is a k−ideal of R if and only if ¯I = I∪I ^ is an ideal of the extension ring ¯ R, called an extension ideal of I.

Theorem 2-7(Chun, Kim and Kim [2]). If R is a k−semiring and ¯ R is the extension ring of R, then each ideal of ¯ R is the extension ideal of a k−ideal of R and each k−ideal of R is the intersection of its extension ideal and R.

Definition 2-8(Chun, Kim and Kim [3]). A mapping f from a k-semiring R into a k-semiring S is called a homo- morphism if f(a+b) = f(a)+f(b) and f(ab) = f(a)f(b) for all a, b∈R.

Theorem 2-9(Chun, Kim and Kim [3]). Let f : R→S be a k-semiring homomorphism. Let ¯ R and ¯ S be extension

rings of R and S respectively. Define a map ¯ f : ¯ R→ ¯ S by f(x) = ¯

 f(x) if x ∈ R f(x ^ ) ^ if x ∈ R ^ .

Then ¯ f is a ring homomorphism, called an extension ring homomorphism of f

Definition 2-10(Kim and Park [5]). A fuzzy ideal of a semiring R is a function A : R→[0, 1] satisfying the fol- lowing conditions:

(1) A(x + y) ≥ min{A(x), A(y)} for all x, y∈R, (2) A(xy) ≥ max{A(x), A(y)} for all x, y∈R.

Lemma 2-11(Chun, Kim and Kim [5]). Let A be a fuzzy ideal of a semiring R. Then A(x)≤A(0) for all x∈R.

Definition 2-12(Chun, Kim and Kim [2]). A k− ideal I of a k−semiring R is maximal provided that I = R and whenever J is a k−ideal of R with I ⊂ J  R then I = J.

Theorem 2-13(Chun, Kim and Kim [2]). Let ¯ R be the extension ring of a commutative k− semiring with identity, I a k−ideal of R and ¯I the extension ideal of I in ¯ R. Then I is a maximal k−ideal of R iff ¯I is a maximal ideal of ¯ R.

Definition 2-14(Kim and Park [5]). Let A be a fuzzy ideal of a semiring R. Then A is called a k−fuzzy ideal of R if A(x + y) = A(0) and A(y) = A(0) imply A(x) = A(0).

Definition 2-15(Kim and Park [5]). Let A be a fuzzy subset of a semiring R. Then the set A t = {x∈R|A(x)≥t}(t∈[0, 1]) is called a level subset of R with respect to A.

Theorem 2-16(Kim and Park [5]). Let A be a fuzzy ideal of a semiring R. Then the level set A t (t ≤ A(0)) is the ideal of R.

Theorem 2-17(Kim and Park [5]). Let A be a fuzzy ideal of a semiring R. If A t is a k−ideal of R for each t(≤ A(0)), then A is a k−fuzzy ideal of R.

However, the converse of Theorem 2-17 does not hold by the following example.

Let R = Z ^∗ , the set of nonnegative integers. Define a fuzzy subset A of R by

A(x) =

⎧ ⎪

⎨

⎪ ⎩

1 if x∈(2)

1 2 if x∈(2, 3) − (2)

0 if x∈Z ^∗ − (2, 3).

Then A is a k−fuzzy ideal but A

= {x∈Z ^∗ |A(x)≥ ¹ ₂ } = (2, 3) is not a k−ideal of R.

However, if A is a k-fuzzy ideal of R, then A R = {x ∈ R|A(x) = A(0)} is also a k-ideal of R.

In general, It is not true that if A is a fuzzy ideal of a semiring R, then A t (t ≤ A(0)) is a k−ideal of R, for we have the following example.

Example 2-18(Kim and Park [5]). Let R = Z ^∗ , the set of nonnegative integers and let I=(2,3) be an ideal of R generated by 2 and 3. Define a fuzzy subset A of R by

A(x) =

 1 if x ∈ I 0 if x /∈ I.

Then A is a fuzzy ideal but A R = I is not a k−ideal of R.

Definition 2-19(Malik and Mordeson [[16]). Let f : R → S be a homomorphism of semirings and B a fuzzy subset of S. We define a fuzzy subset f ⁻¹ B of R by f ⁻¹ B(x) = B(f(x)) for all x∈R.

Theorem 2-20(Kim and Park [5]). Let f : R → S be an epimorphism of semirings and B a fuzzy ideal of S.

Then B is a k− fuzzy ideal of S iff f ⁻¹ (B) is a k− fuzzy ideal of R.

Definition 2-21(Zhang Yue [21]). Let f : R → S be a homomorphism of semirings and A a fuzzy subset of R.

We define a fuzzy subset f(A) of S by f(A)(y) =



Sup {A(t)|t∈R, f(t) = y} iff ⁻¹ (y) = ∅

0 if f ⁻¹ (y) = ∅.

Definition 2-22(Rajesh Kumar [9]). Let R and S be any sets and let f : R → S be a function. A fuzzy subset A of R is called an f− invariant if f(x) = f(y) implies A(x) = A(y), where x, y∈R.

Theorem 2-23(Kim and Park [5]). Let f : R → S be an epimorphism of semirings and A an f− invariant fuzzy ideal of R. Then f(A) is a fuzzy ideal of S.

Theorem 2-24(Kim and Park [5]). Let f and A be as in Theorem 2-23 . Then A is a k− fuzzy ideal of R iff f(A) is a k−fuzzy ideal of S.

Definition 2-25(Liu [15]). A fuzzy ideal of a ring R is a function A : R→[0, 1] satisfying the following axioms

(1) A(x + y)≥min{A(x), A(y)} for all x, y∈R, (2) A(xy)≥max{A(x), A(y)} for all x, y∈R,

(3) A(−x) = A(x) for all x∈R.

Let R be a commutative k−semiring, ¯ R its extension ring. If A is a fuzzy ideal of R such that all its level subsets are k−ideals of R, then R = ∪

t∈ Im A A t , ¯ R = ∪

t∈ Im A

A ¯ t and s > t if and only if A s ⊂A t if and only if ¯ A s ⊂ ¯ A t . Thus we have the following theorem.

Theorem 2-26(Kim and Park [5]). Let R be a commu- tative k−semiring, ¯ R its extension ring. Let A be a fuzzy ideal of R such that all its level subsets are k−ideals of R.

Define the fuzzy subset ¯ A of ¯ R by for all x∈ ¯ R, ¯ A(x) = sup {t|x ∈ ¯ A t , t∈ImA}. Then ¯ A is a fuzzy ideal of ¯ R.

Theorem 2-27(Kim and Park [5]). Let A be as in The- orem 2-26. Then ¯ A is an extension of A.

Definition 2-28(Kim and Park [5]). Let A be a fuzzy ideal of R such that all its level subsets are k−ideals of R.

A fuzzy ideal A of R is called fuzzy maximal if (i)A(0) = 1; (ii)A(e) < A(0); and (iii) whenever A(b) < A(0) for some b ∈ R, then ¯ A(e R ⊕ (rb) ^ ) = A(0) for some r ∈ R, where e R is identity of R.

Kumar[11] defined the fuzzy maximal ideal of a ring as follow;

Definition 2-29(Rajesh Kumar [11]). A fuzzy ideal A of a ring R is called fuzzy maximal if (i)A(0) = 1; (ii)A(e R ) < A(0); and (iii) whenever A(b) < A(0) for some b ∈ R , then A(e R − rb) = A(0) for some r ∈ R.

In the following lemma, we have the relation between the fuzzy maximal k−ideal of a k−semiring and the fuzzy maximal ideal of a ring.

Theorem 2-30(Kim [4]). Let A be as in Definition 2- 28, ¯ A its extension and ¯ R extension ring of R. Then A is a fuzzy maximal ideal of R iff ¯ A is a fuzzy maximal ideal of R. ¯

Definition 2-31. If I is a k−ideal of a commutative semiring R with identity. Then the nil radical of I is de- fined as √

I = {x ∈ R| x ⁿ ∈ I for some n ∈ Z ⁺ }.

Theorem 2-32. If I is a k−ideal of a commutative k−semiring R with identity, then √

I is a k−ideal of R containing I.

Proof. Let x and y be any two elements of √ I, then x ⁿ ∈ I, y ^m ∈ I for some n, m ∈ Z ⁺ . So (x + y) ^n+m ∈ I and (xy) ^n+m ∈ I. Further, for r ∈ R and x ∈ √

I, we have (xr) ⁿ ∈ I, which implies xr ∈ √

I. A similar arguments give that rx ∈ √

I. Thus √

I is a ideal of R. To show that

√ I is a k−ideal of R, we must show that √

¯I∩ R = √ I.

For this, let x be any element of √

¯I ∩ R. Then x ∈ √

¯I

and x ∈ R. So x ^k ∈ ¯I for some k ∈ Z ⁺ . Thus x ^k ∈ I.

Hence x ∈ √

I. Conversely if x is any element of √ I, where x ∈ R, then x ^l ∈ I ⊂ ¯I for some l ∈ Z ⁺ . Thus x ∈ √

¯I ∩ R. Hence √

¯I ∩ R = √

I. Therefore √ I is a k−ideal of R containing I by Theorem 2-7 and √

¯I = √ I.

3. The fuzzy nil radical of a fuzzy ideal

In this section, we define and study a fuzzy nil radical of a fuzzy ideal in a commutative k− semiring with iden- tity. Throughout this paper, unless otherwise all semirings are commutative k−semirings with identity.

Definition 3-1. If α : R→[0, 1] is a fuzzy ideal of a k−semiring R, then the fuzzy set √

α : R→[0, 1] defined

as √

α(x) = 

{α(x ⁿ )|n > 0}

is called the fuzzy nil radical of α.

Example 3-2. Let Z ^∗ denote the semiring of nonnega- tive integers, p be any prime in Z ^∗ and α be a fuzzy subset of Z ^∗ defined by

α(x) =

 1 if x∈ < p ⁿ >,

n−i+1 n−i if x∈ < p ⁿ⁻ⁱ > − < p ⁿ⁻ⁱ⁺¹ >, where i = 1, 2, · · · , n and < p ⁰ >= Z ^∗ . Then α is a fuzzy ideal of Z ^∗ and the fuzzy nil radical of α is given by

√ α(x) =

 1 if x∈ < p >, 0 if x∈ Z ^∗ − < p > .

By similar arguments of Proposition 7-2 of [8], we have the following proposition.

Proposition 3-3. If α : R→[0, 1] is a fuzzy ideal of a k−semiring R, then √ α is a fuzzy ideal of R.

The following theorem is a direct consequence of Def- inition 3-1.

Theorem 3-4. If α : R→[0, 1] and β : R→[0, 1] are fuzzy ideals of a k−semirings R, then the following hold :

(1) √ α = √ α.

(2) If α ⊆ β, then √ α ⊆ √ (3) √ β.

α ∩ β = √ α ∩ √ β.

Definition 3-5(Kumbhojkar and Bapat [6]). Let α : X → L be a fuzzy subset of a set X, where L is a com- plete lattice. α is said to have the supremum property, if

for every subset S of X, there exists x 0 ∈ S such that {α(x)| x ∈ S} = α(x 0 ).

As direct consequences of the definitions, we have the following propositions.

Proposition 3-6. Let f : R → S be a homomorphism of k−semirings, and let α : R → [0, 1] and β : S → [0, 1]

are fuzzy ideals of R and S respectively.

(1) If α has the supremum property, then f(α) has the supremum property.

(2) If β has the supremum property, then f ⁻¹ (β) has the supremum property.

Proposition 3-7. Let 0 ≤ a ≤ 1 and let α : R → [0, 1]

be a fuzzy ideal of a k−semiring R with the supremum property. Then

( √

α) a = √ α a .

Theorem 3-8. If α : R→[0, 1] is a fuzzy ideal of a k−semiring R such that all its level subsets are k−ideals of R and if α has the supremum property, then √

α is a k−fuzzy ideal of R such that all its level subsets are k−ideals of R.

Proof. For t ∈ [0, 1], by Proposition 3-7 ( √

α) t = √ α t . (1)

Since α t is k−ideal of R, ( √ α) t is k−ideal of R by Theo- rem 2-32 and (1). Thus √

α is a k−fuzzy ideal of R such that all its level subsets are k−ideals of R by Theorem 2- 17. This completes the proof.

Theorem 3-9. If α : R → [0, 1] is a fuzzy ideal of a k−semiring R such that all its level subsets are k−ideals of R and if α has the supremum property, then √

α = √

¯α.

Proof. By the definition of the fuzzy nil radical, we have

α ≤ √ α ≤ √

¯α.

So √ α ≤ √

¯α = √

¯α. On the other hand, let x ∈ ¯ R. If x ∈ R, then

√ ¯α(x) =

{¯α(x ⁿ )| n > 0}

=

{α(x ⁿ )| n > 0}

= √ α(x) ≤ √ α(x).

Let x ∈ R ^ . Then x = y ^ for some y ∈ R. So

√ ¯α(x) = √

¯α(y ^ )

= 

{¯α((y ^ ) ⁿ )| n > 0}

= ( 

{¯α((y ^ ) ⁿ )| n > 0, n is even})

∨ ( 

{¯α((y ^ ) ⁿ )| n > 0, n is odd})

= ( 

{α(y ⁿ )| n > 0, n is even})

∨ ( 

{¯α((y ^ ) ⁿ )| n > 0, n is odd })

= ( 

{α(y ⁿ )| n > 0, n is even})

∨ ( 

{α(y ⁿ )| n > 0, n is odd})

= 

{α(y ⁿ )| n > 0}

= √

α(y) = √

α(y) = √

α(y ^ ) = √ α(x).

This completes the proof.

By similar arguments of the Propositions 7-5, 7-6 and 7-7 of [7], we have the following propositions.

Proposition 3-10. Let f : R → S be an epimorphism of k−semirings and let α : R → [0, 1] be a fuzzy ideal of R. Then

(1) f( √ α)   f(α),

(2) If α is constant on Ker f, then f( √

α) =  f(α).

Proposition 3-11. Let f : R → S be a homomorphism of k−semirings and let β : S → [0, 1] be a fuzzy ideal of S. Then

f ⁻¹ (  β) = 

f ⁻¹ (β).

Proposition 3-12. Let α : R → [0, 1] and β : R → [0, 1] are fuzzy ideals of a k−semiring R, then the follow- ing hold :

(1) √

α = √ α,

(2) If α  β, then √ α  √ β, (3) √ α ∩ β = √ α ∩ √ β.

Next, we investigate the properties of the image and the preimage of an extension ideal of a fuzzy ideal under a ho- momorphism of a k−semiring.

Theorem 3-13. Let f : R → S be an epimorphism of k−simirings, and ¯ R and ¯ S extensions of R and S respec- tively. Let ¯ f be the extension of f. Then

(1) β is a fuzzy ideal of S such that all its level subsets are k−ideals of S iff f ⁻¹ (β) is a fuzzy ideal of R such that all its level subsets are k−ideals of R.

(2) ¯ f ⁻¹ ( ¯β) = f ⁻¹ (β)

Proof. (1) See Lemma 3.6 of [4]

(2) If x ∈ ¯ R, then

( ¯ f ⁻¹ ( ¯β))(x) = ¯β ¯ f(x).

If x ∈ R, then ¯β ¯ f(x) = ¯βf(x) = β(f(x)) = (f ⁻¹ β)(x) = f ⁻¹ (β)(x). If x ∈ R ^ , then x = y ^ for some y ∈ R and so ¯β ¯ f(x) = ¯β ¯ f(y ^ ) = ¯β(f(y) ^ ) = ¯β(f(y)) = βf(y) = (f ⁻¹ (β))(y) = f ⁻¹ (β)(y ^ ) = f ⁻¹ (β)(x). In any case, we have ¯ f ⁻¹ ( ¯β) = f ⁻¹ (β). This completes the proof.

Corollary 3-14. Let f, ¯ f, ¯ R, ¯ S, β and ¯β be as in The- orem 3-13. If β is a fuzzy ideal of S with the supremum property. Then

(1) ¯ f ⁻¹ ( 

¯β) = f ⁻¹ ( √ β).

(2) 

( ¯ f) ⁻¹ ( ¯β) = f ⁻¹ ( √ β).

Proof. It is obvious by Theorem 3-9, Proposition 3-11 and Theorem 3-13.

Theorem 3-15. Let f, ¯ f, ¯ R and ¯ S be as in Theorem 3-13 and let α be an f−invariant fuzzy ideal of R. Then

(1) α is a fuzzy ideal of R such that all its level subsets are k−ideals of R iff f(α) is a fuzzy ideal of S such that all its level subsets are k−ideals of S.

(2) f(α) = ¯ f(¯α).

Proof. (1) See Lemma 3.9 of [4].

(2) Let y ∈ ¯ S. We have two cases : (i) y ∈ S and (ii) y ∈ S ^ . (i) If y ∈ S, then there exists a ∈ R such that f(a) = y, since f is an epimorphism. Since f(¯α)(y) = Sup {¯α(t)| ¯ ¯ f(t) = y, t ∈ ¯ R}, ¯ f(t) = y = f(a) = ¯ f(a) and ¯α is ¯ f−invariant(See Lemma 5-6 of [4]), so that ¯ f(¯α)(y) = ¯α(a) = α(a). And f(α)(y) = f(α)(y) = Sup {α(t)| f(t) = y, t ∈ R } = α(a) , since α is f−invariant. (ii) If y ∈ S ^ , then y = z ^ for some z ∈ S. Since f is an epimorphism, there exists b ∈ R such that f(b) = z, and so y = z ^ = f(b) ^ = ¯ f(b ^ ).

So ¯ f(¯α)(y) = ¯ f(¯α)(z ^ ) = Sup {¯α(t)| ¯ f(t) = z ^ , t ∈ R} = ¯α(b ¯ ^ ) = ¯α(b) = α(b), since ¯α is ¯ f−invariant.

And f(α)(y) = f(α)(z ^ ) = f(α)(z) = f(α)(z) = Sup {α(t)| f(t) = z, t ∈ R} = α(b), since α is f−invariant. In any case, we have ¯ f(¯α) = f(α). This completes the proof.

Corollary 3-16. Let f, ¯ f, ¯ R and ¯ S be as in Theorem 3-13. Let α be an f−invariant fuzzy ideal of R with the supremum property. If α is constant on Kerf, then

(1) ¯ f( √

¯α) = f( √ α).

(2)  ¯f(¯α) = f(√α).

Proof. It is obvious by Theorem 3-9, Proposition 3-10

and Theorem 3-15.

Now, we study the relations between the fuzzy nil radi- cal of a fuzzy ideal and the fuzzy primary(prime, maximal) ideal of a k−semiring.

Definition 3-17. Let α : R → [0, 1] be a fuzzy ideal of a k−semiring R. If α is non-constant and for all x, y ∈ R, α(xy) = α(x) or α(y ⁿ ) for some positive inte- ger n, then α is called the fuzzy primary ideal of R.

Definition 3-18. Let α : R → [0, 1] be a fuzzy ideal of a k−semiring R. If ∀a, b ∈ R, either α(ab) = α(a) or α(ab) = α(b), then α is called the fuzzy prime ideal of R.

Theorem 3-19. If α : R → [0, 1] is a fuzzy primary ideal of a k−semiring R, then √

α is a fuzzy prime ideal of R.

Proof. By definition of the fuzzy nil radical ideal,

√ α(xy) = 

{α((xy) ⁿ )| n > 0}

= 

{α(x ⁿ y ⁿ )| n > 0}.

Since α is a fuzzy primary ideal of R,

√ α(xy) = 

{α(x ⁿ )| n > 0} = √ α(x) or

√ α(xy) = 

{α(y ⁿ ) ^m | n > 0, m ∈ Z ⁺ }

= √

α(y ^m )

= √

α(y),

since √ α(y ^m ) =

{α((y ^m ) ⁿ )| n > 0} ≤

{α(y ⁿ )| n >

0} = √

α(y) and √

α(y ^m ) ≥ √

α(y). Thus √

α is a fuzzy prime ideal of R.

Theorem 3-20. Let α be a fuzzy ideal of a k−semiring R such that all its level subsets are k−ideals of R and let α have the supremum property. If √

α is a fuzzy maximal ideal of R, then α is a fuzzy primary ideal of R.

Proof. By Theorem 2-30 and Theorem 3-9, √ α is a fuzzy maximal ideal of R iff √

α is a fuzzy maximal ideal of ¯ R iff √

¯α is a fuzzy maximal ideal of ¯ R. Thus by Theo- rem 5-4 of [13], ¯α is a fuzzy primary ideal of ¯ R and thus α is a fuzzy primary ideal of R.

Malik and Mordeson [17] and Zahedi [23] have defined the radical of a fuzzy ideal J of a ring R containing J. This leads to :

Definition 3-21. Let α : R → [0, 1] be a fuzzy ideal of a k−semiring R and let

r(α) = ∩{β| α ⊆ β, β is fuzzy prime ideal of R}.

Then r(α) is called the fuzzy prime radical of α.

Theorem 3-22. Let α : R → [0, 1] be a fuzzy ideal of a k−semiring R such that all its level subsets are k−ideals of R and let α have the supremum property. Then

(1) √

α = r(α) and r(α) is a k−fuzzy ideal of R such that all its level subsets are k−ideals of R.

(2) r(¯α) = r(α).

Proof. (1) It is obvious by the similar arguments of Theorem 7-14 of [8] and Theorem 3-8.

(2) By (1), √ α = r(α). Since √ α = √

¯α by Theorem 3-9, we have √

¯α = r(α). Also since √

¯α = r(¯α), we have r(¯α) = r(α).

Acknowledgement. I express my thanks to the referees for their valuable comments and suggestions.

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Chang Bum Kim

Professor of Kookmin University

Research Area : Fuzzy Mathematics, Fuzzy Algebra E-mail : [email protected]

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