Structures of Fuzzy Ideals of Γ-Ring

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Malaysian Mathematical Sciences Society

http://math.usm.my/bulletin

Structures of Fuzzy Ideals of Γ-Ring

T.K. Dutta and T. Chanda

Department of Pure Mathamatics, University of Calcutta, 35, Ballygunge Circular Road, Kolkata – 700019, India

Abstract. In this paper we define some compositions of fuzzy ideals in a Γ- ring and study the structures of the set of fuzzy ideals of a Γ-ring. Also we characterize Γ-field, Noetherian Γ-ring, etc. with the help of fuzzy ideals via operator rings of Γ-rings.

2000 Mathematics Subject Classification: 16D30, 16P99

Key words and phrases: Γ-ring, fuzzy left (right) ideal, left (right) operator ring, Notherian Γ-ring.

1. Introduction

The notion of fuzzy ideals in a Γ-ring was introduced by Jun and Lee in [6]. They studied some preliminary properties of fuzzy ideals of Γ-rings. Later in [5] Jun and Hong defined normalized fuzzy ideals and fuzzy maximal ideals in Γ-rings and studied them. In Section 3 of this paper we define some compositions of fuzzy ideals of a Γ-ring and study the structures of the set of fuzzy ideals of a Γ-ring. We show thatF LI(M), the set of all fuzzy left ideals of a Γ-ringM, is a zerosumfree hemiring having infinite element 1, under the operations of sum and composition of fuzzy left ideals. Similar results hold for the set of fuzzy right ideals and that of fuzzy ideals of M. In Section 4 we define a correspondence between the set of all fuzzy ideals of a Γ-ring and the set of all fuzzy ideals of the operator rings of the Γ-ring. We obtain that the lattice of all left (resp. right, two sided) fuzzy ideals is isomorphic to the lattice of all left (resp. right, two sided) fuzzy ideals of the operator ring of the Γ-ring. Using these results we characterize Γ-field, Noetherian Γ-ring etc.

2. Preliminaries

Definition 2.1. [1]LetM andΓbe two additive abelian groups. M is called aΓ-ring if the following conditions are satisfied for all a, b, c∈M and for allα, β, γ∈Γ:

(i) aαb∈M,

(ii) (a+b)αc=aαc+bαc,a(α+β)b=aαb+aβb,aα(b+c) =aαb+aαcand (iii) aα(bβc) = (aαb)βc.

Received:December 13, 2001;Revised: January 6, 2004.

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Definition 2.2. [9]A subsetAof M is called a left (resp. right) ideal ofM ifA is an additive subgroup of M andmαa∈A ( resp. aαm∈A) for all m∈M, α∈Γ anda∈A. If A is both a left and a right ideal of M, then A is called a two sided ideal of M or simply an ideal of M.

Definition 2.3. [9] Let M be a Γ-ring and F the free abelian group generated by Γ×M. ThenA={P

in_i(γ_i, x_i)∈F:a∈M ⇒Pn_iaγ_ix_i= 0}is a subgroup ofF. LetR=F/A, the factor group ofF byA. Let us denote the coset(γ, x)+Aby[γ, x].

It can be verified that[α, x] + [β, x] = [α+β, x], and[α, x] + [α, y] = [α, x+y]for all α, β∈Γ and x, y∈M. We define a multiplication in R by P

i[α_i, x_i]P

j[β_j, y_j] = P

i,j[αi, xiβjyj]. Then Rforms a ring. If we define composition on M×R intoM by aP

i[αi, xi] =P

iaαixi fora∈M,P

i[αi, xi]∈R, thenM is a rightR-module, and we callR the right operator ring of the Γ-ring M. Similarly, we can construct a left operator ring L of M so that M is a left L-module. For subsets N ⊆ M, φ⊆Γ, we denote by [φ, N] the set of all finite sumsP

i[γi, xi] inR, where γi ∈φ, x_i ∈N and we denote by [(Φ, N)] the set of all elements [φ, x] in R where φ∈Φ, x∈N. Thus in particular, R= [Γ, M] and L= [M,Γ]. If there exists an element P

i[δ_i, e_i] ∈ R such that P

ixδ_ie_i =x for every element xof M, then it is called right unity of M. It can be verified that P

i[δ_i, e_i] is the unity of R. Similarly we can define the left unityP

j[f_j, γ_j]which is the unity of the left operator ring L.

Definition 2.4. [6]A nonempty fuzzy subset µ( i.e.,µ(x)6= 0 for somex∈M) of aΓ-ringM is called a fuzzy left(resp. right) ideal ofM if, (i)µ(x−y)≥µ(x)∧µ(y), (ii) µ(xαy)≥µ(y)(resp. µ(xαy)≥µ(x)) for allx, y∈M, and allα∈Γ.

Definition 2.5. AΓ-ringM is said to be commutative ifaγb=bγafor alla, b∈M and for allγ∈Γ.

Definition 2.6. [3]A commutativeΓ-ringM is called aΓ-field if for every non-zero element a of M and for every pair of nonzero elements γ1, γ2 ∈Γ, there exists an elementa⁰ inM such thataγ₁a⁰γ₂b=b for allb∈M.

Definition 2.7. [4] A hemiring [resp. semiring] is a nonempty set R on which operations of addition and multiplication have been defined such that the following conditions are satisfied:

(1) (R,+) is a commutative monoid with identity element 0;

(2) (R, .)is a semigroup [resp. monoid with identity element 1R];

(3) Multiplication distributes over addition from either side;

(4) 0r= 0 =r0 for allr∈R;

(5) 1_R6= 0.

A hemiringR is said to be zerosumfree iff r+r⁰ = 0 implies thatr=r⁰= 0 for all r, r⁰∈R. An elementa of a hemiringR is infinite iffa+r=afor allr∈R.

3. Operations on fuzzy ideals

Throughout this paper M denotes a Γ-ring with left unity and right unity and F LI(M) (resp. F RI(M), F I(M)) denotes the set of all fuzzy left ideals (resp.

fuzzy right ideals, fuzzy ideals) ofM. Also we assume that for any fuzzy left (resp.

right, two sided) idealσofM,σ(0M) = 1.

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Definition 3.1. Let µ, σ be two fuzzy subsets ofM. Then the sum µ⊕σ, product µΓσ and compositionµ◦σof µandσ are defined as follows:

(µ⊕σ)(x) =

(sup_x=u+v[min[µ(u), σ(v)]] foru, v ∈M

0 otherwise.

(µΓσ)(x) =

(sup_x=uγv[min[µ(u), σ(v)]] foru, v∈M andγ∈Γ

0 otherwise.

(µ◦σ)(x) =











sup[mini[min[µ(ui), σ(vi)]]], 1≤i≤n, x=

n

P

i=1

uiγivi, u_i, v_i∈M, γ_i∈Γ

0 otherwise.

Proposition 3.1. Let µ, σ be two fuzzy ideals ofM. ThenµΓσ⊆µ◦σ⊆µ∩σ.

Proof. From the definitions ofµΓσandµ_◦σ, it follows thatµΓσ⊆µ_◦σ. Letx∈M andx=

n

P

i=1

uiγivi,ui,vi∈M,γi∈Γ for i= 1,2, . . . , n. Now

µ(x) = µ(

n

X

i=1

u_iγ_iv_i)

≥ min{µ(u1γ1v1), µ(u2γ2v2), . . . , µ(unγnvn)}

≥ min{µ(u₁), µ(u₂), . . . , µ(u_n)}.

Similarly

σ(x)≥min{σ(v1), σ(v₂), . . . , σ(v_n)}.

Thus

(µ∩σ)(x) = min{µ(x), σ(x)} ≥min

i [min[µ(ui), σ(vi)]].

So (µ∩σ)(x)≥sup[mini[min[µ(ui), σ(vi)]]], 1≤i ≤n, x=

n

P

i=1

uiγivi,ui, vi ∈M, γ_i∈Γ = (µ◦σ)(x). Also ifµ◦σ(x) = 0, thenµ◦σ(x)≤µ∩σ(x). Soµ◦σ⊆µ∩σ.

ThusµΓσ⊆µ◦σ⊆µ∩σ.

Proposition 3.2. Letµ₁,µ₂∈F LI(M)[resp. F RI(M),F I(M)]. Thenµ₁⊕µ₂∈ F LI(M)[resp. F RI(M),F I(M)].

Proof. Letx, y∈M andγ ∈Γ. Also let (µ₁⊕µ₂)(y)>(µ₁⊕µ₂)(x). Then there exist p, q ∈ M such that y = p+q and for any u, v ∈ M, for which x = u+v, min[µ1(p), µ2(q)] >min[µ1(u), µ2(v)]. Let u, v ∈M be such thatx=u+v. Now x−y= (u−p) + (v−q). So

(µ1⊕µ2)(x−y)≥min[µ1(u−p), µ2(v−q)]

≥min[min[µ1(u), µ1(p)],min[µ2(v), µ2(q)]]

= min[min[µ₁(u), µ₂(v)],min[µ₁(p), µ₂(q)]]

= min[µ1(u), µ2(v)].

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So

(µ₁⊕µ₂)(x−y)≥ sup

x=u+v

[min[µ₁(u), µ₂(v)]], u, v∈M

= (µ₁⊕µ₂)(x)

= min[(µ₁⊕µ₂)(x),(µ₁⊕µ₂)(y)].

Similarly we can show that (µ1⊕µ2)(x−y)≥min[(µ1⊕µ2)(x),(µ1⊕µ2)(y)], in all other cases. Again, let y = p+q, p, q ∈ M. Then xγy = xγp+xγq, x∈ M andγ∈Γ. Now (µ1⊕µ2)(xγy)≥min[µ1(xγp),µ2(xγq)]≥min[µ1(p), µ2(q)]. Thus (µ1⊕µ2)(xγy)≥sup_y=p+q[min[µ1(p), µ2(q)]],p, q∈M = (µ1⊕µ2)(y). Lastly, since µ₁(0_M) =µ₂(0_M) = 1, (µ₁⊕µ₂)(0_M) = 1. Soµ₁⊕µ₂∈F LI(M).

Proposition 3.3. Let µ, σ, δ∈F LI(M) [resp. F RI(M),F I(M)]. Then (i) µ⊕σ=σ⊕µ,

(ii) (µ⊕σ)⊕δ=µ⊕(σ⊕δ), (iii) µ⊆µ⊕σ,

(iv) ifµ⊆σ, then µ⊕δ⊆σ⊕δ, (v) µ⊕µ=µ,

(vi) θ⊕µ=µ=µ⊕θ whereθ(∈F LI(M))is defined by θ(x) =

(1 if x= 0M, x∈M 0 if x6= 0M.

Proof. The proof is a routine matter of verification and so we omit it.

Proposition 3.4. Let µ, σ ∈ F LI(M) [resp. F RI(M), F I(M)]. Then µ◦σ ∈ F LI(M)[resp. F RI(M),F I(M)].

Proof. The proof is similar to the proof of the Proposition 3.2 and so we omit it.

Proposition 3.5. Let µ, σ, δ∈F LI(M)[resp. F RI(M),F I(M)]. ThenµΓσ⊆δ iffµ◦σ⊆δ.

Proof. Ifµ◦σ⊆δ, thenµΓσ⊆µ◦σ⊆δ. Conversely, letµΓσ⊆δ. Letx∈M be such thatx=

n

P

i=1

uiγivi, ui, vi∈M,γi∈Γ for 1≤i≤n. Now δ(x) =δ(

n

X

i=1

uiγivi)

≥min[δ(u1γ1v1), δ(u2γ2v2), . . . , δ(unγnvn)]

≥min[(µΓσ)(u₁γ₁v₁),(µΓσ)(u₂γ₂v₂), . . . ,(µΓσ)(u_nγ_nv_n)]

≥min[min[µ(u1), σ(v1)], . . . ,min[µ(un), σ(vn)]].

So

δ(x)≥ sup

x=

n

P

i=1

u_iγ_iv_i

[min

i [min[µ(ui), µ(vi)]]] = (µ◦σ)(x).

Also if (µ_◦σ)(x) = 0, then µ◦σ(x) ≤δ(x). Thus µ◦σ⊆δ.

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Proposition 3.6. Let µ, σ, δ∈F LI(M) [resp. F RI(M),F I(M)]. Then (i) ifµ⊆σ, then µ◦δ⊆σ◦δ,

(ii) (µ◦σ)◦δ=µ◦(σ◦δ),

(iii) µ◦σ=σ◦µ ifM is commutative,

(iv) 1◦µ= µ where 1 ∈ F LI(M) is defined by 1(x) = 1 for all x∈ M [resp.

µ◦1 =µ,1◦µ=µ◦1 =µ].

Proof. The proof of (i) to (iii) follows from the definitions of compositions of fuzzy ideals and so we omit it. (iv) AsM is with left unityP

j

[fj, γj]∈Lwhich is defined byP

j

fjγjx=xfor every elementxinM, it follows form definition that 1_◦µ=µ.

Similarly we can prove the following proposition.

Proposition 3.7. Let µ, σ, δ∈F LI(M) [resp. F RI(M),F I(M)]. Then (i) µ◦(σ⊕δ) =µ◦σ⊕µ◦δ,

(ii) (σ⊕δ)◦µ=σ◦µ⊕δ◦µ.

Theorem 3.1. Let M be a Γ-ring. Then F LI(M) [resp. F RI(M), F I(M)] is a zerosumfree hemiring(resp. hemiring, semiring) having infinite element 1 under the operations of sum and composition of fuzzy left ideals.

Proof. From the Propositions 3.2, 3.3, 3.4, 3.6 and 3.7, it follows that F LI(M) is a hemiring under the operations of sum and composition of fuzzy left ideals. Now (1⊕µ)(x) = sup_x=u+v[min[1(u), µ(v)]]≥min[1(x), µ(0_M)] = 1(x)≥(1⊕µ)(x) for all x∈M. So 1⊕µ= 1 for allµ∈F LI(M). Thus 1 is an infinite element ofF LI(M).

Lastly we assume thatµ⊕σ=θ forµ, σ∈F LI(M). Thenµ⊆µ⊕σ=θ⊆µ.So µ=θ. SoF LI(M) is zerosumfree. Hence the theorem.

Lemma 3.1. [6] Intersection of a nonempty collection of fuzzy left ideals (resp.

fuzzy right ideals, fuzzy ideals) is a fuzzy left ideal (resp. fuzzy right ideal, fuzzy ideal) of M.

Theorem 3.2. F LI(M)[resp. F RI(M),F I(M)] is a complete lattice.

Proof. We define a relation ‘≤’ on F LI(M) as follows µ₁ ≤ µ₂ iff µ₁(x) ≤µ₂(x) for all x∈M. Then F LI(M) is a poset w.r.t. ‘≤’. Now 1∈F LI(M) andµ≤1 for allµ ∈ F LI(M). So 1 is the greatest element of F LI(M). Let{µ_i, i∈ I} be a nonempty family of fuzzy left ideals of M. Then by Lemma 3.1, it follows that

∩_i∈Iµi ∈ F LI(M). Also it is the glb of {µi|i ∈ I}. Consequently F LI(M) is a

complete lattice.

4. Corresponding fuzzy ideals

Throughout this paper R denotes the right operator ring and L denotes the left operator ring ofM.

Definition 4.1. For a fuzzy subset µ of R, we define a fuzzy subset µ^∗ of M by µ^∗(a) = inf_γ∈Γµ([γ, a]) wherea∈M. For a fuzzy subsetσ ofM, we define a fuzzy subsetσ^∗⁰ ofR byσ^∗⁰(P

i

[α_i, a_i]) = inf_m∈Mσ(P

i

mα_ia_i) whereP

i

[α_i, a_i]∈R.

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Definition 4.2. For a fuzzy subset δ of L, we define a fuzzy subset δ⁺ of M by δ⁺(a) = infγ∈Γδ([a, γ]), wherea∈M. For a fuzzy subsetη ofM, we define a fuzzy subsetη⁺⁰ ofL byη⁺⁰(P

i

[a_i, α_i]) = inf_m∈Mη(P

i

a_iα_im)whereP

i

[a_i, α_i]∈L.

Lemma 4.1. If {µi|i∈I}is a collection of fuzzy subsets ofR, then

∩_i∈Iµ^∗_i = (∩_i∈Iµi)^∗. Proof. Letx∈M. Now

(∩i∈Iµi)^∗(x) = inf

γ∈Γ[(∩i∈Iµi)([γ, x])]

= inf

γ∈Γ[inf

i∈I(µi[γ, x])]

= inf

i∈I[ inf

γ∈Γ[µ_i([γ, x])]]

= inf

i∈I[µ^∗_i(x)]

= (∩_i∈Iµ^∗_i)(x).

So∩_i∈Iµ^∗_i = (∩_i∈Iµi)^∗.

Proposition 4.1. Ifµ∈F I(R)[resp. F RI(R),F LI(R)], then µ^∗∈F I(M)[resp.

F RI(M),F LI(M)].

Proof. Letµbe a fuzzy ideal ofR. Thenµ(0_R) = 1. Now µ^∗(0_M) = inf

γ∈Γµ([γ,0_M]) = inf

γ∈Γµ(0_R) = 1.

Soµ^∗ is nonempty. Leta, b∈M andα∈Γ. Now µ^∗(a−b) = inf

γ∈Γµ([γ, a−b])

= inf

γ∈Γµ([γ, a]−[γ, b])

≥min[ inf

γ∈Γµ([γ, a])],inf

γ∈Γµ([γ, b])]

= min[µ^∗(a), µ^∗(b)].

Again

µ^∗(aαb) = inf

γ∈Γµ([γ, aαb]) = inf

γ∈Γµ([γ, a][α, b])≥ inf

γ∈Γµ([γ, a]) =µ^∗(a).

Again

µ^∗(aαb) = inf

γ∈Γµ([γ, aαb])

= inf

γ∈Γµ([γ, a][α, b])

≥ inf

γ∈Γµ([α, b])

=µ([α, b])

≥ inf

γ∈Γµ([γ, b]) =µ^∗(b).

Soµ^∗ is a fuzzy ideal ofM.

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Proposition 4.2. If σ ∈ F I(M) [resp. F RI(M), F LI(M)], then σ^∗⁰ ∈ F I(R) [resp. F RI(R),F LI(R)].

Proof. Let σ be a fuzzy ideal of M. Then σ(0_M) = 1. Now σ^∗⁰([γ,0_M]) = infm∈Mσ(mγ0M) =σ(0M) = 1. So σ^∗⁰ is nonempty. Let P

i

[αi, ai],P

j

[βj, bj]∈R.

Then σ^∗⁰(X

i

[α_i, a_i]−X

j

[β_j, b_j]) = inf

m∈Mσ(X

i

mα_ia_i−X

j

mβ_jb_j)

≥ inf

m∈M[min[σ(X

i

mα_ia_i), σ(X

j

mβ_jb_j)]]

= min[ inf

m∈Mσ(X

i

mαiai), inf

m∈Mσ(X

j

mβjbj)]

= min[σ^∗⁰(X

i

[αi, ai]), σ^∗⁰(X

j

[βj, bj])].

Again, σ^∗⁰(X

i

[α_i, a_i]X

j

[β_j, b_j]) = σ^∗⁰(X

i,j

[α_i, a_iβ_jb_j]

= inf

m∈Mσ(X

i,j

mα_ia_iβ_jb_j)

≥ inf

m∈M[min

i [σ(mα1(X

j

a1βjbj)), σ(mα2(X

j

a2βjbj)), . . .]]

≥ inf

m∈M[min

i [σ(X

j

a1βjbj), σ(X

j

a2βjbj), . . .]]

= min[σ(X

j

a1βjbj), σ(X

j

a2βjbj), . . .]

≥ inf

m∈M[σ(X

j

mβjbj)]

= σ^∗⁰(X

j

[β_j, b_j].

Similarly we can show that σ^∗⁰(P

i

[α_i, a_i]P

j

[β_j, b_j]) ≥ σ^∗⁰(P

i

[α_i, a_i]). So σ^∗⁰ is a

fuzzy ideal ofR.

Similarly we can prove the following Proposition.

Proposition 4.3. If δ∈F I(L) [resp. F RI(L),F LI(L)], thenδ⁺∈F I(M)[resp.

F RI(M),F LI(M)].

Proposition 4.4. Ifη∈F I(M)[resp. F RI(M),F I(M)], thenη⁺⁰ ∈F I(L)[resp.

F RI(L),F LI(L)].

Theorem 4.1. The lattices of all fuzzy ideals (resp. fuzzy left ideals ) of M and R are isomorphic via the inclusion preserving bijection σ→σ^∗⁰ whereσ∈F I(M) [resp. F LI(M)] andσ^∗⁰ ∈F I(R) [resp. F LI(R)].

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Proof. First we shall show that (σ^∗⁰)^∗=σ, whereσ∈F I(M). Let a∈M. Then (σ^∗⁰)^∗(a) = inf

γ∈Γ[σ^∗⁰([γ, a])]

= inf

γ∈Γ[ inf

m∈M[σ(mγa)]]

≥ inf

γ∈Γ[ inf

m∈M[σ(a)]] =σ(a).

Soσ⊆(σ^∗⁰)^∗. LetP

i

[e_i, δ_i] be the left unity ofM. ThenP

i

e_iδ_ix=xfor allx∈M. Now

σ(a) =σ(X

i

eiδia)

≥min

i [σ(e₁δ₁a), σ(e₂δ₂a), . . .])]]

≥ inf

γ∈Γ[ inf

m∈M[σ(mγa)]] = (σ^∗⁰)^∗(a).

So (σ^∗⁰)^∗⊆σ. Henceσ= (σ^∗⁰)^∗. Again, letµ∈F I(R). Now (µ^∗)^∗⁰(X

k

[αk, ak]) = inf

m∈M[µ^∗(X

k

mαkak)]

= inf

m∈M[ inf

γ∈Γ[µ(γ,X

k

mαkak)]]

= inf

m∈M[ inf

γ∈Γ[µ([γ, m]X

k

[α_k, a_k])]]

≥ µ(X

k

[α_k, a_k]).

Soµ⊆(µ^∗)^∗⁰. LetP

j

[δ_j⁰, e⁰_j] be the right unity of M. Then µ(X

k

[αk, ak]) = µ(X

j

[δ_j⁰, e⁰_j]X

k

[αk, ak])

≥ min

j [µ([δ₁⁰, e⁰₁]X

k

[α_k, a_k]), µ([δ⁰₂, e⁰₂]X

k

[α_k, a_k]), . . .]

≥ inf

m∈M[ inf

γ∈Γ[µ([γ, m]X

k

[αk, ak])]]

= (µ^∗)^∗⁰(X

k

[αk, ak]).

Soµ ⊇(µ^∗)^∗⁰. Thusµ = (µ^∗)^∗⁰. Thus the correspondence σ→ σ^∗⁰ is a bijection.

Now letσ1, σ2∈F I(M) be such that σ1⊆σ2. Then σ₁^∗⁰(X

i

[αi, ai]) = inf

m∈Mσ1(X

i

mαiai)

≤ inf

m∈Mσ2(X

i

mαiai) =σ₂^∗⁰(X

i

[αi, ai])

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for all P

i

[αi, ai]∈R. So σ₁^∗⁰ ⊆σ₂^∗⁰. Similarly we can show that ifµ1 ⊆µ2, where µ1, µ2∈F I(R), thenµ^∗₁⊆µ^∗₂. So the mappingσ→σ^∗⁰ is a lattice isomorphism.

Similarly we can prove the following theorem.

Theorem 4.2. The lattices of all fuzzy ideals (resp. fuzzy right ideals) ofM and L are isomorphic via the inclusion preserving bijection η→η⁺⁰, where η ∈F I(M) [resp. F RI(M)] andη⁺⁰ ∈F I(L) [resp. F RI(L)].

Theorem 3.2 maybe obtained as a corollary of the above theorems.

Corollary 4.1. F RI(M)[resp. F I(M),F LI(M)] is a complete lattice.

Proof. The corollary follows from the above theorem and the facts that F I(M),

F RI(M) andF LI(M) are complete lattices.

Theorem 4.3. A commutativeΓ–ringM is aΓ– field if and only if for every fuzzy idealσ ofM,σ(x) =σ(y)< σ(0M) for allx, y∈M\{0M}.

Proof. Let σ be a fuzzy ideal of M and σ(x) = σ(y) < σ(0M) for all x, y ∈ M\{0M}. Let P

i

[αi, ai],P

j

[βj, bj] ∈ R\{0R}. Then there exist m, m⁰ in M such thatP

i

mαiai6= 0M andP

j

m⁰βjbj6= 0M. Now σ^∗⁰(X

i

[α_i, a_i]) = inf

m∈Mσ(X

i

mα_ia_i) = inf

m∈Mσ(X

j

mβ_jb_j) =σ^∗⁰(X

j

[β_j, b_j]) (sinceσ(x) =σ(y)< σ(0M) for allx, y∈M\{0M}). So

σ^∗⁰(X

i

[αi, ai]) =σ^∗⁰(X

j

[βj, bj])< σ^∗⁰(0R) for allP

i

[αi, ai],P

j

[βj, bj]∈R\{0R}. Letµbe a fuzzy ideal ofR and X

i

[α_i, a_i],X

j

[β_j, b_j]∈R\{0R}.

Then µ(X

i

[αi, ai]) = (µ^∗)^∗⁰(X

i

[αi, ai]) = (µ^∗)^∗⁰(X

j

[βj, bj]) =µ(X

j

[βj, bj])< µ(0R).

Also it follows from Lemma 3.4 of [2] that R is commutative. Consequently by Proposition 3.1.10 of [7] it follows that R is a field and hence M is a Γ-field ([2, Theorem 3.5]).

Conversely, suppose that M is a Γ-field and x, y ∈ M\{0_M}. Then there exist γ1, γ2∈Γ such that [γ1, x]6= 0R and [γ2, y]6= 0R. Letσbe fuzzy ideal ofM. Then σ^∗⁰ is a fuzzy ideal of R. Since M is a Γ-field, R is a field. So, by the Proposition 3.1.10 of [7] it follows that

σ^∗⁰(X

i

[αi, ai]) =σ^∗⁰(X

j

[βj, bj])< σ^∗⁰(0R)

(10)

for allP

i

[αi, ai],P

j

[βj, bj]∈R\{0R}. Now σ(x) = (σ^∗⁰)^∗(x) = inf

γ∈Γσ^∗⁰([γ, x]) = inf

γ∈Γσ^∗⁰([γ, y]) = (σ^∗⁰)^∗(y) =σ(y)< σ(0M).

Soσ(x)) =σ(y)< σ(0M) for allx, y∈M\{0M}.

Definition 4.3. A commutative Γ-ring M is said to be Noetherian if for every ascending chain I1 ⊆I2 ⊆I3 ⊆. . . of ideals of M there exists a positive integer n such that Im=In for all m≥n.

Theorem 4.4. A commutative Γ-ring M is Noetherian if every fuzzy ideal of M has finite values.

Proof. LetM be a commutative Γ-ring and every fuzzy ideal ofM have finite values.

Let σbe a fuzzy ideal of M. Then since σ^∗⁰(P

i

[γi, ai]) = inf_m∈Mσ(P

i

mγiai), σ^∗⁰ is of finite values wheneverσ is of finite values. Letµ be a fuzzy ideal ofR. Since for every fuzzy ideal µof R, µ = (µ^∗)^∗⁰, it follows thatµ has finite values. So By Theorem 7 of [10] it follows thatR is Noetherian and henceM is Noetherian.

Acknowledgment. We are thankful to the referee for his valuable suggestions.

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