# On Krull's intersection theorem of fuzzy ideals

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PII. S0161171203013097 http://ijmms.hindawi.com © Hindawi Publishing Corp.

### ON KRULL’S INTERSECTION THEOREM OF FUZZY IDEALS

V. MURALI and B. B. MAKAMBA

We deal with Krull’s intersection theorem on the ideals of a commutative Noether-ian ring in the fuzzy setting. We ﬁrst characterise products of ﬁnitely generated fuzzy ideals in terms of fuzzy points. Then, we study the question of uniqueness and existence of primary decompositions of fuzzy ideals. Finally, we use such de-compositions and a form of Nakayama’s lemma to prove the Krull intersection theorem. Fuzzy-points method on ﬁnitely generated fuzzy ideals plays a central role in the proofs.

2000 Mathematics Subject Classiﬁcation: 13A15, 03E72, 13C12.

1. Introduction. After the ﬁrst paper on fuzzy groups in 1971 by Rosen-feld, a decade passed before researchers started to look more closely into the notions extending fuzzy subsets to groups, rings, vector spaces, and other algebraic objects, [2, 3,8,9,10]. There were attempts to unify these studies in one coherent way in the form of modules, [4, 5, 6]. The notion of prime ideal was generalized to fuzzy prime ideals of a ring, and thus initiating the study of radicals and primary ideals in the fuzzy case by Malik and Mordeson [7, 8, 9, 10, 11]. In this paper, we study the concepts of primary decompo-sitions of fuzzy ideals and the radicals of such ideals over a commutative ring. Using such decompositions and a form of Nakayama’s lemma, we prove Krull’s intersection theorem on fuzzy ideals. We cleverly use the idea of fuzzy points belonging to fuzzy subsets to work out a number of proofs, creating a new way of extending ideas (see, e.g.,Proposition 3.3, Theorems3.5and3.11, Proposition 5.2, and so on). (Even though these results are known in the liter-ature, our proofs are new and are based on the fuzzy-point concept.)

We brieﬂy describe the framework of the paper. InSection 2, we introduce notions from fuzzy set theory such as fuzzy ideals, fuzzy points, sums and products on fuzzy ideals, and ﬁnitely generated fuzzy ideals. InSection 3, we recall deﬁnitions of prime, primary ideals, and their radicals from [11]. Then, we state and prove two uniqueness theorems of primary decompositions. In Section 4, we introduce irreducible fuzzy ideals to establish the existence the-orem of primary decomposition. Section 5deals with the Krull intersection theorem, namelyn=1µn=0 under suitable conditions onµ.

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Nakayama’s lemma states that ifMis a ﬁnitely generated module overRand ifIis an ideal contained in the Jacobson radical ofRwithM=IM, thenM=0. A consequence of this fact is that ifIis an ideal of a commutative Noetherian ringRcontained in the Jacobson radical ofR, then∩∞

n=1In=0. In the literature,

this is one of the forms in which Krull’s intersection theorem is stated. In the following, we are interested in translating these results in the fuzzy set theory setting. Firstly, we summarise the basic theory of fuzzy ideals in order to set the foundation leading up to decomposition theorems of fuzzy ideals. We use [0,1], the real unit interval, as a chain with the usual ordering in which stands for inﬁmum (or intersection) andstands for supremum (or union). IfR is treated as a set, then any mappingµ:R→[0,1]is known as afuzzy subset ofR. The pointwise ordering of the power setIRinduces the notions of containment, intersection, and union among the fuzzy subsets ofR in a natural way, viz.µ≤ν if and only ifµ(r )≤ν(r )for allr∈R. In particular, the top element isχR, which we denote byR. The bottom element isχ0, which

we denote by 0. Afuzzy subgroup ofRis a mappingµ:R→[0,1]such that µ(r )=µ(−r )for allr∈Randµ(r+s)≥µ(r )∧µ(s)for allr ,s∈R. A fuzzy subsetµ:R→[0,1]is afuzzy idealofRif (i)µis a fuzzy subgroup ofRand (ii), for each pairr ,s∈R, we haveµ(r s)≥µ(r )∨µ(s). WhenR is a ﬁeld, (ii) becomesµ(r s)=µ(s)for allr ∈R\ {0},s∈R. Suppose thatM is a module overRandµis a fuzzy subgroup ofM; then,µis afuzzy submoduleofMif, for eachr∈Randm∈M, we haveµ(r m)≥µ(m). Lett∈[0,µ(0)]where the 0 inµ(0)denotes the additive identity inR, andµis a fuzzy ideal ofR. The subsetµt= {r∈R:µ(r )≥t}is called thet-level idealofR. It is clear thatµis a fuzzy ideal ofRif and only if eachµt, 0tµ(0), is an ideal ofR. Without loss of generality, throughout this paper,we tacitly assume that µ(0)=1for every fuzzy idealµof R.

Leta∈R, 0< λ≤1. Afuzzy pointaλofRis a fuzzy subsetaλ:R[0,1] deﬁned by for allr ∈R, (r )=λ if r =aand 0 if r a. The collection of all fuzzy points ofR is denoted byFP(R). Byµ, we meanµ(a)λ, and∈sµmeansµ(a) > λ. The collection of fuzzy points ofµis denoted by

FP(µ). The pointais called thesupportofandλis its height. More generally, for a fuzzy subsetµofR, thesupportofµis the set suppµ= {x∈R:µ(x) >0}. Ifµis a fuzzy ideal then suppµis an ideal ofR.

The operations ofsums and products on fuzzy subsetsµ and ν ofR are deﬁned as follows:

(µ+ν)(x)=supµx1

∧νx2

:x=x1+x2, x1,x2∈R

, (µ◦ν)(x)=supµ(r )∧ν(s):x=r s, r ,s∈R,

(µν)(x)=sup

  

n

i=1

µri∧νsi:x= n

i=1

risi, ri,si∈R, n∈N

  .

(2.1)

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For later use, we state without proof that the product distributes over addition for fuzzy ideals, viz. µ(ν+ω)= µν+µω, see [9, Proposition 2.2.8]. Also, µkµr=µk+r andk)r=µkrfor allk,rN. Ifaλ1

1 ,...,aλnn arenfuzzy points

ofR, then the fuzzy idealµ, generated by1

1 ,...,aλnn, is deﬁned as n

i=1χRaλii using the above deﬁnitions of sums and products. Such a µ is denoted by

1,aλ22,...,aλnnand is called afinitely generated fuzzy ideal. The following

two propositions deal with the product of fuzzy ideals generated by fuzzy points the proofs of which are found in [11].

Proposition 2.1. The fuzzy ideal 1122 = a1λ122 for 11,aλ22

FP(R).

A consequence ofProposition 2.1is the following proposition.

Proposition2.2. If ν=aλ11,aλ22, thenν2=(a211,aλ1

1 22,(a222, where

ν2means the productνν.

Generally, ifν= aλ1

1 ,aλ22,...,a

λk

k for somek∈N, thenνn= {(ar111···

(ark

k)λk:r1+r2+···+rk=n}, where we assume thatλi=1 whenri=0.

Note2.3. We observe that the support suppµof a ﬁnitely generated fuzzy idealµ(ﬁnitely generated fuzzy submoduleµ) is ﬁnitely generated. Further,µ is ﬁnite-valued. Conversely, we have the following proposition.

Proposition2.4. IfR is a Noetherian ring, then every finite-valued fuzzy ideal is finitely generated.

Proof. See [6, Proposition 3.9].

Ifωis a fuzzy ideal of R, there are two equivalent ways in which we can deﬁne the radical. Given a fuzzy idealω, the nil radical ofis deﬁned by Rω(x)={ω(xn):nN},xR, and the prime radical ofωis deﬁned as

:ω≤ν, νis a prime fuzzy ideal ofR}. That they are indeed equivalent is shown in [11]. We denote either of the radical ofωby√ω. The radical√ωis a fuzzy ideal ofRsuch thatω≤√ω. A fuzzy idealωinRis a maximal fuzzy ideal ifω≤ν,ν a fuzzy ideal ofR, impliesν=R orω1=ν1[14]. A fuzzy

idealωis a maximal fuzzy ideal ofRif and only if Im(ω)= {1,t}for some t∈[0,1)andω1is a maximal ideal ofR. The fuzzy Jacobson radical ofR is

the inﬁmum (intersection) of all maximal fuzzy ideals ofR.

In the paragraph precedingProposition 2.1, we deﬁned the product of two fuzzy subsetsµandν. If one of them is a fuzzy point and the other is a fuzzy ideal, the product can be viewed as follows: let µ be a fuzzy ideal ofR and FP(R). The productrλµ is deﬁned byrλµ(x)=n

i=1µ(yi):x= n

i=1r yi}, forx∈R. Ifx has no such decomposition, then rλµ(x)=0. We

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In terms of fuzzy pointsµ:R={rλFP(R):rλRµ}. In this paper, we only have occasion to use the residualµ:whereaλis the fuzzy ideal generated by.

3. Primary decomposition. In this section, we describe a method of decom-posing a fuzzy ideal as an intersection of primary fuzzy ideals. Assuming the existence of such a decomposition, we prove some uniqueness results. Most of the results of this section are well known in the literature [11,12], albeit proved using a diﬀerent technique from ours. We ﬁrst recall the deﬁnitions of prime and primary fuzzy ideals.

A fuzzy idealµ is called a prime(primary) fuzzy ideal inR ifµR and, for,sλFP(R),rβsλµimplies eitherrβµorsλµ (sλµ), and is calledν-primary if, ﬁrstly,µis primary and, secondly,ν=√µ.

Definition3.1. Let{νi:i=1,2,...,n}be a collection of prime fuzzy ideals and{µi:i=1,2,...,n}a ﬁnite collection ofνi-primary fuzzy ideals ofR. Then, µ=ni=1µiis called aprimary decompositionofµ.

Note3.2. This decomposition is said to bereduced or irredundant, (we use the two terms interchangeably) if (1) theν12,...,νn are all distinct and

(2)µj≥ni=1, ijµi, for allj=1,2,...,n.

Proposition3.3. Letµ be aν-primary fuzzy ideal ofRandrλ∈FP(R). Then, the following are satisfied:

(i) if µ, thenµ:rλ =R;

(ii) if µ, thenµ:rλis aν-primary fuzzy ideal ofR. This implies, in particular,ν=µ:.

Proof. (i) Letrλ∈µ. For convenience, throughout the proof, ωdenotes

the fuzzy ideal generated by, that is,ω= rλ. Then, forrR,r1ωµ.

That is,r1µ:ωfor allrR. Therefore,µ:rλ =R. (ii) Letrλ∈µ. Consider 1

1 r2λ2 ∈µ :ω. That is,r1λ1r2λ2ω≤µ. Since µ is

primary, either1

1 ∈√µorr2λ2∈√µ.

Let >0. Then, there exist positive integers n1 and n2 such that either

(rn1

1 1−∈µor(r

n2

2 2−∈µ. Therefore,(r

n1

1 1−ω≤µor(r

n2

2 2−ω≤µ.

This impliesr1λ1−∈√µ:ωor r2λ2−∈√µ:ω. As >0 is arbitrary, either

1

1 ∈√µ:ωorr

λ2

2 ∈√µ:ω. This proves thatµ:is primary.

We now claim thatµ: =µ=ν.

Letµ. For >0, there exists a positive integernsuch that(sn)β−µ. Sinceω≤R,(sn)β−µ:ω. Asis arbitrary,sβµ:ω. Therefore,µ

µ:. The reverse inclusion follows from a similar argument. Therefore, ν=µ:.

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(i) ωcontainsωjfor somej, (ii) ω≥ni=1ωi,

(iii) ω≥ni=1ωi.

Proof. The implications (i)(ii)(iii) are straightforward. To show that

(iii)(i), assume thatω≥ni=1ωi. Suppose thatωdoes not containωjfor all

j=1,2,...,n. Then, for eachj, there existsrλj

j ∈ωj\ω. Therefore,r1λ1r2λ2···

rλn

n ∈ωsinceωis prime. Butr1λ1r2λ2···rnλn∈ n

to our assumption. This completes the proof.

Theorem 3.5. Letµ,µ12,...,µn be a set ofn+1fuzzy ideals of R such thatµ=µ1∧µ2∧···∧µnis a reduced primary decomposition ofµ. Letωbe a

prime fuzzy ideal ofR. Then, the following statements are equivalent: (i) ω=νifor someiwhereνi=√µi,

(ii) there exists a fuzzy pointrλRsuch thatµ:rλis aω-primary fuzzy ideal,

(iii) there exists a fuzzy pointrλRsuch thatω=µ:rλ.

Proof. (i)(ii). There exists a fuzzy pointrλ∈nj=1µj\µi since the

de-composition is reduced. It is easily shown that(nj=1µj):rλ =(µj:rλ).

ByProposition 3.3,µi:is aνi-primary fuzzy ideal andµj:rλ =Rfor ji. Hence,µ: =n

j=1(µj:rλ)=µi:since others areR. Therefore,

µ:isω-primary becauseω=νi. (ii)(iii) is clear.

(iii)(i). LetRsuch thatω=µ:rλ. Then,ω=n i=1{

µi:such thatR}since ifrλµithenµi:rλ =Ror ifrλµithenω=n

i=1νiby Proposition 3.3. Now, byProposition 3.4,ω≥νjfor somejsinceω=nj=1νj.

Therefore,ω=νjfor somej.

Theorem3.6(the ﬁrst uniqueness theorem). Suppose that a fuzzy idealµ has a reduced primary decomposition in the formµ=µ1∧µ2∧ ··· ∧µn with

νi=√µifori=1,...,n. Also, suppose thatµ=η1∧η2∧···∧ηmwithξj= √ηj

forj=1,...,mis another reduced primary decomposition ofµ. Then,n=m and{ν12,...,νn} = {ξ12,...,ξm}.

Proof. Firstly, we observe thatξ1is a prime fuzzy ideal ofR. Setξ1.

Consider a fuzzy point(m

j=2ηj)\η1. ByProposition 3.3, η1:is a

ξ1-primary fuzzy ideal ofRand ξ1=

η1:. Sincerλ∈ m

j=2ηj, we have

ηj: =R, for allj=2,3,...,m. Therefore,µ:rλ =η1:rλis aω-primary ideal ofR. Thus, we have a fuzzy point R such thatω=η

1:rλ =

µ:.

Secondly, byTheorem 3.5, there exists ani∈ {1,2,...,n}such thatω=νi. That is, ξ1=νi for some 1≤i≤n. Repeating the process, we get ξj =νk

for some 1≤k≤nfor eachj =2,...,m. This implies that12,...,ξm} ⊆

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the reverse inclusion of the above subsets, implyingn≤m. This completes the proof.

Definition3.7. The set12,...,νn}, as stated in the above theorem, is known as the set ofassociated primeideals ofµand is denoted by AssR(µ). In this case, we say that eachνibelongs toµ.

For the next proposition, we require the notion of minimal prime fuzzy ideal. We recall that a prime fuzzy idealµcontainingνis said to beminimalifν≤µ and, whenever a primeωsatisﬁes the inequality ν≤ω≤µ, thenω=ν or ω=µ.

Proposition3.8. Letµbe a fuzzy ideal with a primary decomposition, and letνbe a prime fuzzy ideal such thatν≥µ. Then,νis a minimal prime fuzzy ideal containingµif and only ifνis a minimal member ofAssR(µ).

Proof. Letµ=ni=1µibe a reduced primary decomposition ofµ. Letνi= µifori=1,2,...,nbe the associated prime ideals ofµ. We note thatνµif

and only ifν=√ν≥√µ. Now,√µ=ni=1√µi= n

i=1νi. So,ν≥µif and only if

ν≥ni=1νi. Therefore,ν≥νjfor somejbetween 1 andnbyProposition 3.4.

(). Firstly, we observe thatνj≥µbecauseνj= √µj≥µj≥µ. This is true for allj=1,2,...,n. From above,ν≥νj for somej between 1 andn. Now, minimality ofνimpliesν=νj. Therefore,νis a minimal member of AssR(µ). ().ν≥µimplies that there exists a minimal prime fuzzy idealωsuch that ν≥ω≥µ. Therefore, there exists aνj∈AssR(µ)such thatν≥ω≥νj≥µ as stated above. Minimality ofνin AssR(µ)implies thatν=ω=νjwhich, in turn, implies thatνis a minimal prime fuzzy ideal containingµ.

Note3.9. Minimal members of AssR(µ)are called theminimalorisolated primes ofµ. The other members of AssR(µ)are calledembedded primes ofµ.

Lemma 3.10. Letω be aν-primary fuzzy ideal of R. Ifaλ∈ν, thenω: =ω.

Proof. Clearly,ω:aλ ≥ω. Let rβ∈ω:. Then,rβaλ∈ω. But, by

hypothesis,ν=ω. Therefore,rβωsinceωisν-primary. This com-pletes the proof.

Theorem3.11(the second uniqueness theorem). Letµbe a fuzzy ideal of R. Suppose thatµhas the following two reduced primary decompositions:

µ=µ1∧µ2∧···∧µn, withνi=√µifori=1,...,n,

µ=ξ1∧ξ2∧···∧ξn, withνi=

ξifori=1,...,n. (3.1)

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Proof. Forn=1, AssR(µ)= {ν1}. So,µ11. Forn >1, letµibe a minimal

member of AssR(µ). Then, there exists a fuzzy pointn

j=1, jiνj\νi. (Oth-erwise,νi≥nj=1, jiνjwhich, in turn, will imply thatνi> νjfor somej, con-tradicting the minimality ofνi.) Now, for >0,rλ−νiandrλ−n

j=1, jiνj. So, we can assume without loss of generality thatis strictly inn

j=1, jiνj\νi. Thus, there exists a positive integermjsuch that(rmj)λµjfor everyji. Sett≥max{mj:j=1,2,...,n, ji}. Then,(rt)λµjand(rt)λνi. There-fore,µ:(rt)λ =(µj):(rt)λ =µi:(rt)λsinceµj:(rt)λ =Rforji. Now,µi:(rt)λ =µi byLemma 3.10 since µi is assumed to beνi-primary. Similarly,ξi=µi:(rt)λfor a large enough value fort. Thus, we haveµi=ξi.

4. Irreducible decompositions. Throughout this section and the next one, Rdenotes a Noetherian commutative ring with unity.

Definition4.1. Letµ be a fuzzy ideal ofR. It is said to beirreducibleif µRand wheneverµ=µ1∧µ2, whereµ1andµ2are fuzzy ideals ofR, then

µ=µ1orµ=µ2.

Proposition4.2. Letµ be an irreducible fuzzy ideal ofR. Then,Im(µ)= {1,t}for some0≤t <1, and the top cutµ1is irreducible.

Proof. Suppose thatµ(z)=t1< t2=µ(y)1 for somez,y∈R. Deﬁne

µ1(x)=         

1, x∈µ1,

µ(x), x∈µt21,

t2 otherwise,

µ2(x)=   

1, x∈µt2,

µ(x), otherwise.

(4.1)

Firstly, we note thatµ12≥µandµ1≠µ,µ2≠µ. Secondly,

µ1∧µ2(x)=   

1, x∈µ1,

µ(x), otherwise =µ(x).

(4.2)

This contradicts the irreducibility ofµ. Therefore,|Im(µ)| =2.

Further, suppose thatµ1is reducible. Then, there exist two proper idealsJ 1

andJ2ofRsuch thatµ1=J1∩J2. Fori=1,2, let

µi(x)=

  

1, x∈Ji,

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We note thatµi≥µandµiµfori=1,2,

µ1∧µ2

(x)=

  

1, x∈µ1,

µ(x) otherwise =µ(x).

(4.4)

This is a contradiction to the irreducibility ofµ. Therefore,µ1is irreducible.

With each irreducible crisp ideal ofR, there is an associated irreducible fuzzy ideal ofRas the following proposition shows.

Proposition4.3. LetIbe an irreducible ideal ofRand define a fuzzy ideal µasµ(x)=1ifx∈Iandαifx∈Ifor some0≤α <1. Thenµis irreducible.

Proof. Suppose that there exist fuzzy ideals µ12µ such that µ = µ1∧µ2. This implies the existence of anx∈Rsuch thatµ1(x) > µ(x).

Simi-larly, there is ay∈Rsuch thatµ2(y) > µ(y). Consider the ideals generated by

Iandx,(I,x), andIandy,(I,y). Clearly,Iis contained inI,x∩I,y, and also, I is distinct from each of I,xand I,y. Let a∈ I,x ∩ I,y, thena=r1x1+s1y=r2x2+s2x, wherex1,x2∈I andr1,r2,s1,s2∈R. Now,

µ(r1x1−r2x2)=1=µ(s2x−s1y). This implies thatµ(r1x1)=µ(r2x2)and

µ(s2x)=µ(s1y). Also,µ1(s2x)≥µ1(x) > µ(x)≥αand µ2(s2x)=µ2(s2x−

s1y+s1y)≥µ2(s2x−s1y)∧µ2(s1y)=µ2(s1y)≥µ2(y) > µ(y)≥α.

There-fore,1∧µ2)(s2x) > αwhich implies thatµ(s2x) > α. Hence,µ(s2x)=1. This

implies thata∈Iand soI,y ∩ I,x ⊆I. Therefore,I= I,y ∩ I,x. This contradicts the irreducibility ofI.

Corollary4.4. Ifµis a fuzzy ideal such thatµ1is irreducible andIm(µ)= {1,α}, thenµis irreducible.

Note4.5. A prime fuzzy ideal is necessarily irreducible; however, the con-verse need not hold as the following example shows.

Example4.6. LetR be the ringZ+Z. ConsiderI=4Z+Z. We claim thatI is irreducible. Suppose not. Then, there exist two idealsJ1 andJ2such that

JiI;I⊂Jifori=1,2 whereI=J1∩J2. It is an easy exercise to check that

J1andJ2are of the formk1Z+k2Zandm1Z+m2Zfor some positive integers

k1,k2,m1andm2, respectively. Then, 4Z+Z=(k1Z+k2Z)∩(m1Z+m2Z)=

(k1Z∩m1Z)+(k2Z∩m2Z), which implies triviallyk2=1=m2. Further, 4Z

k1Z∩m1Z. Therefore,k1,m1divide 4. Each possible value fork1and m1as

a divisor of 4 leads to an absurd equality. Thus, the claim is proved. Now, we obtain the required example by deﬁningµ(x)to be equal to 1 ifx∈4Z+Zand 1/3 otherwise.

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Proof. Sinceµ is irreducible,µR; Im(µ)= {1,α}for some 0≤α <1 andµ1is irreducible. Now, we need only to show thatµ1is primary. To this

end, considerr s∈µ1andsµ1. SinceRis Noetherian, we can ﬁnd a positive

integernsuch that the ascending chainµ1:r ⊆µ1:r2µ1:r3 ⊆ ···

ends withµ1:rn =µ1:rn+1 = ···.

Claim. The top cutµ1=(µ1+rnR)∩(µ1+Rs).

Clearly,µ11+rnR)∩1+Rs). For the reverse inclusion, consider,y

1+rnR)∩1+Rs). Then,y=s

1+rns2=s3+tsfors1,s3∈µ1,s2∈R, and

t∈R. The elementr y=r s1+rn+1s2=r s3+t(r s)∈µ1, which implies that

rn+1s

2∈µ1, which, in turn, implies thats2∈µ1:rn+1. Therefore,s2∈µ1:

rn, which implies thatrns2µ1. Therefore,yµ1. This proves the claim.

Sinceµ1is irreducible, eitherµ1=µ1+rnRor µ1=µ1+Rs. Buts1+

Rs)\µ1. Hence,µ1=µ1+rnR, implying that rnRµ1. This completes the

proof.

Lemma4.8. Letµbe a fuzzy ideal ofRsuch that|Im(µ)| =2. Then,µ can be expressed as a finite intersection of irreducible fuzzy ideals ofR.

Proof. The top cutµ1is an intersection of a collection of irreducible ideals Ji,i=1,2,...,nofR[13]. Suppose that Im(µ)= {1,t}, 0≤t <1. Deﬁneνi(x)= 1 ifx∈Jiandtotherwise. Eachνiis irreducible andµ=ni=1νi.

Theorem4.9. Letµbe a proper finite-valued fuzzy ideal ofR. Then,µcan be expressed as a finite intersection of irreducible fuzzy ideals ofR.

Proof. Firstly, we observe that|Im(µ)|<∞, so we can assume that Im(µ)=

{1,tn,tn−1,...,t1}, where 1> tn> tn−1>···> t1.

Let µ1 = ∩m1

i=1J1i, µtn = ∩mi=21Jtni,...,µt2 = ∩mi=n1Jt2i where J11,...,Jtn1,...,

Jt21,...,Jt2mn are all irreducible ideals ofR. Deﬁne the fuzzy idealsν’s as fol-lows:

ν1i(x)=   

1, ifx∈J1i,

tn, otherwise,

νtni(x)=   

1, ifx∈Jtni, tn−1, otherwise,

.. .

νt2i(x)=   

1, ifx∈Jt2i,

t1, otherwise.

(4.5)

Then, all the aboveν1i,...,νt2i,...are irreducible. Further,

m1

i=1

ν1i(x)=   

1, ifx∈ ∩m1

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m2

i=1

νtni(x)=   

1, ifx∈ ∩m2

i=1Jtni=µtn, tn−1, otherwise,

.. .

(4.6)

Therefore,µ=(ν1i)∧(νtni)∧···∧(νt2i).

We end this section with the following corollary on the existence of primary decompositions. Theorem 4.9 tells us that every proper ﬁnite-valued fuzzy ideal ofRcan be expressed as a ﬁnite intersection of irreducible fuzzy ideals ofR. Now, fromProposition 4.7, we know that an irreducible fuzzy ideal is primary. Thus, we have the following corollary.

Corollary4.10. Letµbe a proper finite-valued fuzzy ideal ofR. Then,µ has a reduced primary decomposition.

5. Krull’s intersection theorem. In general, it is not true thatn=1µn=0

for a fuzzy idealµofR. But Krull proved that such equality holds under certain additional hypotheses in the crisp case [1]. In this section, we study his theorem in the fuzzy case.

Lemma5.1. Letνbe a fuzzy ideal ofR. Suppose that√νis a finitely gener-ated fuzzy ideal of R. Then, there exists a natural numbernsuch that(√ν)n≤ν.

Proof. Let√ν= aλ11,a2λ2,...,aλkkfor some positive integerk. Given an arbitrary >0, there exists, for eachi between 1 andk, ni∈N such that (ani

i )λi−∈ν. Setn=1+

k

i=1(ni−1). ByProposition 2.2,(√ν)n={(ar111···

(ark

k)λk:r1+r2+···+rk=n}, where we assume thatλi=1 whenri=0. It is

easy to see that anrj≥njfor some 1≤j≤n. Therefore,(arj

j )λj−∈ν. As is arbitrary,(√ν)n≤ν.

Proposition5.2. Letµ be a finite-valued proper fuzzy ideal ofRandνa prime fuzzy ideal ofR. Then,ν∈AssR(µ)if and only if there existsaλFP(R) such thatµ: =ν.

Proof. (). Suppose that there exists aaλ∈FP(R)such thatµ:aλ =ν.

Then,µ:is prime, thereforeν-primary. Then, byTheorem 3.5,ν=νifor some 1≤i≤nsuch thatνi=√µiwhereµ=µ1∧µ2∧ ··· ∧µnis a reduced

primary decomposition, that is,ν∈AssR(µ).

(). Suppose thatν∈AssR(µ)whereµ=µ1∧µ2∧ ··· ∧µn,νi=√µi, is a

reduced primary decomposition. Considerωj=ni=1,ijµi. This implies that µ≤ωj≤µj sinceµ=ni=1µi is a reduced primary decomposition. Now, by Proposition 2.4,νjis ﬁnitely generated. So,Lemma 5.1implies that there exists a positive integertsuch thatνt

j=(√µj)t≤µj. Now,νjtωj≤µjωj≤µj∧ωj= µ. Let ube the least positive integer such that νu

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Choose νu−1

j ωj\µ. Then, ∈ωj\µ. Now, µi :aλ =R for alli= 1,2,...,n,ijandµj:isνj-primary byProposition 3.3. Thus,µ:aλ =

n

i=1(µi:aλ)=µj:. Therefore,µ:is νj-primary. Now, aλνj

νu

jωj≤µ. Hence,νj≤µ:aλ ≤νjsinceµ:aλ =µj:aλ ≤

µj:aλ =νj. This completes the proof.

Theorem5.3. Letµbe a finite-valued fuzzy ideal of R, and setν=∞n=1µn. Thenν=µν.

Proof. The case whenµ=Ris clear. Now, suppose thatµR. Then,µνRsinceµν≤µ. Thus,µνhas a primary decomposition byCorollary 4.10. Let µν 1∧µ2∧ ··· ∧µn with νi =√µi, i=1,2,...,n be a reduced primary

decomposition ofµν. We need only to show that ν ≤µν since the reverse inequality is always true. Suppose that there is a natural numberi, 1≤i≤n, such thatν≤µi. Then, there exists a fuzzy pointν\µi. Now,aλµµν=

n

i=1µi≤µiand µi isνi-primary. Therefore,µ≤√µi=νi sinceaλ∈µi, and

νiis ﬁnitely generated byProposition 2.4. Now, byLemma 5.1, there exists a positive integer t such thatνt

i ≤µi, thus ν=

n=1µn≤µt ≤νit ≤µi. This contradicts our assumption thatν≤µi. Therefore,ν≤µifor alli=1,2,...,n, and hence,ν≤µν. This completes the proof.

Lemma5.4(Nakayama’s lemma). Letµ be a finitely generated fuzzy sub-module of a sub-moduleM over a commutative ringR. Letν be a fuzzy ideal of Rcontained in the fuzzy Jacobson radical ofR. Ifµ=νµ, thenµis the trivial fuzzy submodule.

Note5.5. By the trivial fuzzy submodule ofM, we mean, for all x∈M, µ(x)=1 ifx=0 and 0 ifx≠0.

Proof. Sinceµ=νµ, suppµ=(suppν)(suppµ). SetI=suppνbe the crisp

ideal inR. Let Mbe any maximal ideal of R. Construct ωon R as follows: ω(x)=1 ifx∈M and 0 otherwise. Note thatω is a maximal fuzzy ideal ofR. Now,ν is contained in the fuzzy Jacobson radical ofR which is itself contained inω. Therefore,ν≤ω, implyingI=suppν⊆suppω=M. Thus,I is contained in the crisp Jacobson radical ofR. By Nakayama’s lemma for the crisp case, suppµ=0. This completes the proof.

Theorem5.6(Krull’s intersection theorem). Letµ be a finitely generated fuzzy ideal ofR such thatµ is contained in the fuzzy Jacobson radical ofR. Then∞n=1µn=0.

Proof. Let ν= n=1µn. Since µ is ﬁnitely generated, µ is ﬁnite-valued,

which, in turn, implies thatν is ﬁnite-valued. ByTheorem 5.3, ν=µν. Now, Nakayama’s lemma implies thatν=0.

Acknowledgment. This research was supported by the Foundation for

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V. Murali: Department of Mathematics (Pure & Applied), Rhodes University, Graham-stown 6140, South Africa