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

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

**V. MURALI and B. B. MAKAMBA**

Received 11 May 2001

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, namely*∞ _{n}_{=1}µn_{=}*

_{0 under suitable conditions on}

_{µ}_{.}

Nakayama’s lemma states that if*M*is a ﬁnitely generated module over*R*and
if*I*is an ideal contained in the Jacobson radical of*R*with*M=IM*, then*M=*0.
A consequence of this fact is that if*I*is an ideal of a commutative Noetherian
ring*R*contained in the Jacobson radical of*R*, then*∩∞*

*n=*1*In=*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) and*∨*stands for supremum (or union).
If*R* is treated as a set, then any mapping*µ*:*R→[*0*,*1*]*is known as a*fuzzy*
*subset* of*R*. The pointwise ordering of the power set*IR*_{induces the notions}
of containment, intersection, and union among the fuzzy subsets of*R* in a
natural way, viz.*µ≤ν* if and only if*µ(r )≤ν(r )*for all*r∈R*. In particular,
the top element is*χR*, which we denote by*R. The bottom element isχ*0, which

we denote by 0. A*fuzzy subgroup* of*R*is a mapping*µ*:*R→[*0*,*1*]*such that
*µ(r )=µ(−r )*for all*r∈R*and*µ(r+s)≥µ(r )∧µ(s)*for all*r ,s∈R*. A fuzzy
subset*µ*:*R→[0,1]*is a*fuzzy ideal*of*R*if (i)*µ*is a fuzzy subgroup of*R*and
(ii), for each pair*r ,s∈R*, we have*µ(r s)≥µ(r )∨µ(s)*. When*R* is a ﬁeld, (ii)
becomes*µ(r s)=µ(s)*for all*r* *∈R\ {*0*}*,*s∈R*. Suppose that*M* is a module
over*R*and*µ*is a fuzzy subgroup of*M; then,µ*is a*fuzzy submodule*of*M*if,
for each*r∈R*and*m∈M*, we have*µ(r m)≥µ(m)*. Let*t∈[*0*,µ(*0*)]*where the
0 in*µ(*0*)*denotes the additive identity in*R*, and*µ*is a fuzzy ideal of*R*. The
subset*µt= {r∈R*:*µ(r )≥t}*is called the*t-level ideal*of*R. It is clear thatµ*is
a fuzzy ideal of*R*if and only if each*µt*_{, 0}_{≤}_{t}_{≤}_{µ(}_{0}_{)}_{, is an ideal of}_{R}_{. Without}
loss of generality, throughout this paper,*we tacitly assume that* *µ(*0*)=*1*for*
*every fuzzy idealµof* *R*.

Let*a∈R*, 0*< λ≤*1. A*fuzzy pointaλ*_{of}_{R}_{is a fuzzy subset}_{a}λ_{:}_{R}_{→}_{[}_{0}_{,}_{1}* _{]}*
deﬁned by for all

*r*

*∈R*,

*aλ*

_{(r )}_{=}_{λ}_{if}

_{r}

_{=}_{a}_{and 0 if}

_{r}_{≠}

_{a}_{. The collection}of all fuzzy points of

*R*is denoted byFP

*(R)*. By

*aλ*

_{∈}_{µ}_{, we mean}

_{µ(a)}_{≥}_{λ}_{,}and

*aλ*

_{∈s}_{µ}_{means}

_{µ(a) > λ}_{. The collection of fuzzy points of}

_{µ}_{is denoted by}

FP*(µ)*. The point*a*is called the*support*of*aλ*_{and}_{λ}_{is its height. More generally,}
for a fuzzy subset*µ*of*R*, the*support*of*µ*is the set supp*µ= {x∈R*:*µ(x) >*0*}*.
If*µ*is a fuzzy ideal then supp*µ*is an ideal of*R*.

The operations of*sums* and *products* on fuzzy subsets*µ* and *ν* of*R* are
deﬁned as follows:

*(µ+ν)(x)=*sup*µx*1

*∧νx*2

:*x=x*1*+x*2*, x*1*,x*2*∈R*

*,*
*(µ◦ν)(x)=*sup*µ(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)

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,
*µk _{µ}r_{=}_{µ}k+r*

_{and}

_{(µ}k_{)}r_{=}_{µ}kr_{for all}

_{k,r}_{∈}_{N}

_{. If}

*1*

_{a}λ1 *,...,aλnn* are*n*fuzzy points

of*R*, then the fuzzy ideal*µ*, generated by*aλ*1

1 *,...,aλnn*, is deﬁned as
*n*

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

1*,aλ*22*,...,aλnn*and is called a*finitely 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* *aλ*_{1}1*aλ*_{2}2* = a*_{1}*λ*1*aλ*_{2}2 *for* *aλ*_{1}1*,aλ*_{2}2 *∈*

FP*(R).*

A consequence ofProposition 2.1is the following proposition.

**Proposition _{2.2}.**

_{If}*ν=aλ*

_{1}1

*,aλ*

_{2}2

*2*

_{, then}ν*=(a*2

_{1}

*)λ*1

*,aλ*1

1 *aλ*22*,(a*22*)λ*2*, where*

*ν*2_{means the product}_{νν}_{.}

Generally, if*ν= aλ*1

1 *,aλ*22*,...,a*

*λk*

*k* for some*k∈*N, then*νn= {(ar*11*)λ*1*···*

*(ark*

*k)λk*:*r*1*+r*2*+···+rk=n}, where we assume thatλi=*1 when*ri=*0.

**Note _{2.3}.**

_{We observe that the support supp}

*µ*

_{of a ﬁnitely generated fuzzy}ideal

*µ*(ﬁnitely generated fuzzy submodule

*µ*) is ﬁnitely generated. Further,

*µ*is ﬁnite-valued. Conversely, we have the following proposition.

**Proposition _{2.4}.**

_{If}R

_{is a Noetherian ring, then every finite-valued fuzzy}*ideal is finitely generated.*

**Proof.** _{See [}_{6}_{, Proposition 3.9].}

If*ω*is a fuzzy ideal of *R*, there are two equivalent ways in which we can
deﬁne the radical. Given a fuzzy ideal*ω*, the nil radical of*Rω*is deﬁned by
*Rω(x)={ω(xn _{)}*

_{:}

_{n}_{∈}_{N}

_{},}_{x}_{∈}_{R, and the prime radical of}_{ω}_{is deﬁned as}

*{ν*:*ω≤ν, ν*is a prime fuzzy ideal of*R}*. That they are indeed equivalent is
shown in [11]. We denote either of the radical of*ω*by*√ω*. The radical*√ω*is
a fuzzy ideal of*R*such that*ω≤√ω*. A fuzzy ideal*ω*in*R*is a maximal fuzzy
ideal if*ω≤ν*,*ν* a fuzzy ideal of*R*, implies*ν=R* or*ω*1* _{=}_{ν}*1

_{[}

_{14}

_{]. A fuzzy}

ideal*ω*is a maximal fuzzy ideal of*R*if and only if Im*(ω)= {*1*,t}*for some
*t∈[*0*,*1*)*and*ω*1_{is a maximal ideal of}_{R}_{. The fuzzy Jacobson radical of}_{R}_{is}

the inﬁmum (intersection) of all maximal fuzzy ideals of*R*.

In the paragraph precedingProposition 2.1, we deﬁned the product of two
fuzzy subsets*µ*and*ν*. 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 of*R* and
*rλ _{∈}*

_{FP}

_{(R)}_{. The product}

_{r}λ_{µ}_{is deﬁned by}

_{r}λ_{µ(x)}_{=}_{{λ}_{∧}n*i=*1*µ(yi)*:*x=*
*n*

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

In terms of fuzzy points*µ*:*R={rλ _{∈}*

_{FP}

_{(R)}_{:}

_{r}λ_{R}_{≤}_{µ}}_{. In this paper, we}only have occasion to use the residual

*µ*:

*aλ*

_{}_{where}

_{a}λ_{}_{is the fuzzy ideal}generated by

*aλ*

_{.}

**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 ideal*µ* is called a *prime*(*primary*) fuzzy ideal in*R* if*µ*≠*R* and,
for*rβ _{,s}λ_{∈}*

_{FP}

_{(R)}_{,}

_{r}β_{s}λ_{∈}_{µ}_{implies either}

_{r}β_{∈}_{µ}_{or}

_{s}λ_{∈}_{µ (s}λ_{∈}√_{µ)}_{, and is}called

*ν*-primary if, ﬁrstly,

*µ*is 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 of*νi*-primary fuzzy ideals of*R*. Then,
*µ=ni=1µi*is called a*primary decomposition*of*µ.*

**Note _{3.2}.**

_{This decomposition is said to be}

_{reduced}_{or}

_{irredundant}_{, (we}use the two terms interchangeably) if (1) the

*ν*1

*,ν*2

*,...,νn*are all distinct and

(2)*µj≥ni=*1*, i*≠*jµi*, for all*j=*1*,*2*,...,n*.

**Proposition _{3.3}.**

_{Let}µ*FP*

_{be a}ν_{-primary fuzzy ideal of}R_{and}rλ∈*(R)*

_{.}*Then, the following are satisfied:*

(i) *if* *rλ _{∈}_{µ}_{, then}_{µ}*

_{:}

_{r}λ_{ =}_{R}_{;}(ii) *if* *rλ _{∈}_{µ}_{, then}_{µ}*

_{:}

_{r}λ_{}_{is a}_{ν}_{-primary fuzzy ideal of}_{R}_{. This implies, in}*particular,ν=µ*:

*rλ*

_{}_{.}**Proof.** _{(i) Let}*rλ∈µ. For convenience, throughout the proof,* *ω*_{denotes}

the fuzzy ideal generated by*rλ*_{, that is,}_{ω}_{= r}λ_{}_{. Then, for}_{r}_{∈}_{R}_{,}* _{r}*1

_{ω}_{≤}_{µ}_{.}

That is,*r*1_{∈}_{µ}_{:}_{ω}_{for all}_{r}_{∈}_{R}_{. Therefore,}_{µ}_{:}_{r}λ_{ =}_{R}_{.}
(ii) Let*rλ∈µ. Consider* *rλ*1

1 *r*2*λ*2 *∈µ* :*ω. That is,r*1*λ*1*r*2*λ*2*ω≤µ. Since* *µ* is

primary, either*rλ*1

1 *∈√µ*or*r*2*λ*2*∈√µ*.

Let * >*0. Then, there exist positive integers *n*1 and *n*2 such that either

*(rn*1

1 *)λ*1−*∈µ*or*(r*

*n*2

2 *)λ*2−*∈µ*. Therefore,*(r*

*n*1

1 *)λ*1−*ω≤µ*or*(r*

*n*2

2 *)λ*2−*ω≤µ*.

This implies*r*1*λ*1−*∈√µ*:*ω*or *r*2*λ*2−*∈√µ*:*ω*. As * >*0 is arbitrary, either

*rλ*1

1 *∈√µ*:*ω*or*r*

*λ*2

2 *∈√µ*:*ω*. This proves that*µ*:*rλ*is primary.

We now claim that*µ*:*rλ _{ =}√_{µ}_{=}_{ν}*

_{.}

Let*sβ _{∈}√_{µ}*

_{. For}

_{ >}_{0, there exists a positive integer}

_{n}_{such that}

_{(s}n_{)}β−_{∈}_{µ}_{.}Since

*ω≤R*,

*(sn*

_{)}β−_{∈}_{µ}_{:}

_{ω}_{. As}

_{}_{is arbitrary,}

_{s}β_{∈}√_{µ}_{:}

_{ω}_{. Therefore,}

*√*

_{µ}_{≤}

*µ*:*rλ _{}*

_{. The reverse inclusion follows from a similar argument. Therefore,}

*ν=µ*:

*rλ*

_{}_{.}

(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 that*ω≥ni=*1*ωi*. Suppose that*ω*does not contain*ωj*for all

*j=*1*,*2*,...,n*. Then, for each*j*, there exists*rλj*

*j* *∈ωj\ω*. Therefore,*r*1*λ*1*r*2*λ*2*···*

*rλn*

*n* *∈ω*since*ω*is prime. But*r*1*λ*1*r*2*λ*2*···rnλn∈*
*n*

*j=1ωj*. This is a contradiction

to our assumption. This completes the proof.

**Theorem** **3.5.** *Letµ,µ*_{1}*,µ*_{2}*,...,µn* *be a set ofn+*1*fuzzy ideals of* *R* *such*
*thatµ=µ*1*∧µ*2*∧···∧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 pointrλ _{∈}_{R}_{such that}_{µ}*

_{:}

_{r}λ_{}_{is a}_{ω}_{-primary fuzzy}*ideal,*

(iii) *there exists a fuzzy pointrλ _{∈}_{R}_{such that}_{ω}_{=}_{µ}*

_{:}

_{r}λ_{}_{.}**Proof.** (i)*⇒*(ii). There exists a fuzzy point*rλ∈n _{j}_{=}*

_{1}

*µj\µi*since the

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

ByProposition 3.3,*µi*:*rλ _{}*

_{is a}

_{νi}_{-primary fuzzy ideal and}

_{µj}_{:}

_{r}λ_{ =}_{R}_{for}

*j*≠

*i*. Hence,

*µ*:

*rλ*

_{ =}n*j=*1*(µj*:*rλ)=µi*:*rλ*since others are*R*. Therefore,

*µ*:*rλ _{}*

_{is}

_{ω}_{-primary because}

_{ω}_{=}_{νi}_{.}(ii)

*⇒*(iii) is clear.

(iii)*⇒*(i). Let*rλ _{∈}_{R}*

_{such that}

_{ω}_{=}_{µ}_{:}

_{r}λ_{}_{. Then,}

_{ω}_{=}n*i=1{*

*µi*:*rλ _{}*

_{such}that

*rλ*

_{∈}_{R}}_{since if}

_{r}λ_{∈}_{µi}_{then}

_{µi}_{:}

_{r}λ_{ =}_{R}_{or if}

_{r}λ_{∈}_{µi}_{then}

_{ω}_{=}n*i=*1*νi*by
Proposition 3.3. Now, byProposition 3.4,*ω≥νj*for some*j*since*ω=nj=1νj*.

Therefore,*ω=νj*for some*j*.

**Theorem3.6**(the ﬁrst uniqueness theorem)**.** *Suppose that a fuzzy idealµ*
*has a reduced primary decomposition in the formµ=µ*1*∧µ*2*∧ ··· ∧µn* *with*

*νi=√µifori=*1*,...,n. Also, suppose thatµ=η*1*∧η*2*∧···∧ηmwithξj= √ηj*

*forj=*1,...,m*is another reduced primary decomposition ofµ. Then,n=m*
*and{ν*1*,ν*2*,...,νn} = {ξ*1*,ξ*2*,...,ξm}.*

**Proof.** _{Firstly, we observe that}*ξ*_{1}_{is a prime fuzzy ideal of}*R*_{. Set}*ξ*_{1}*=ω*_{.}

Consider a fuzzy point*rλ _{∈}_{(}m*

*j=2ηj)\η*1. ByProposition 3.3, *η*1:*rλ*is a

*ξ*1-primary fuzzy ideal of*R*and *ξ*1*=*

*η*1:*rλ*. Since*rλ∈*
_{m}

*j=*2*ηj*, we have

*ηj*:*rλ _{ =}_{R}*

_{, for all}

_{j}_{=}_{2}

_{,}_{3}

_{,...,m}_{. Therefore,}

_{µ}_{:}

_{r}λ_{ =}_{η}_{1}

_{:}

_{r}λ_{}_{is a}

_{ω}_{-primary}ideal of

*R*. Thus, we have a fuzzy point

*rλ*

_{∈}_{R}_{such that}

_{ω}_{=}_{η}1:*rλ =*

*µ*:*rλ _{}*

_{.}

Secondly, byTheorem 3.5, there exists an*i∈ {1,2,...,n}*such that*ω=νi*.
That is, *ξ*1*=νi* for some 1*≤i≤n*. Repeating the process, we get *ξj* *=νk*

for some 1*≤k≤n*for each*j* *=*2*,...,m*. This implies that*{ξ*1*,ξ*2*,...,ξm} ⊆*

the reverse inclusion of the above subsets, implying*n≤m*. This completes
the proof.

**Definition _{3.7}.**

_{The set}

*{ν*

_{1}

*,ν*

_{2}

*,...,νn}*

_{, as stated in the above theorem, is}known as the set of

*associated prime*ideals of

*µ*and is denoted by Ass

*R(µ)*. In this case, we say that each

*νi*belongs to

*µ*.

For the next proposition, we require the notion of minimal prime fuzzy ideal.
We recall that a prime fuzzy ideal*µ*containing*ν*is said to be*minimal*if*ν≤µ*
and, whenever a prime*ω*satisﬁes the inequality *ν≤ω≤µ*, then*ω=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 of*Ass*R(µ).*

**Proof.** _{Let}*µ=n _{i}_{=1}µi*

_{be a reduced primary decomposition of}

*µ. Letνi=*

*√*

_{µi}_{for}

_{i}_{=}_{1}

_{,}_{2}

_{,...,n}_{be the associated prime ideals of}

_{µ}_{. We note that}

_{ν}_{≥}_{µ}_{if}

and only if*ν=√ν≥√µ*. Now,*√µ=ni=*1*√µi=*
_{n}

*i=*1*νi*. So,*ν≥µ*if and only if

*ν≥ni=1νi*. Therefore,*ν≥νj*for some*j*between 1 and*n*byProposition 3.4.

(*⇒*). Firstly, we observe that*νj≥µ*because*νj= √µj≥µj≥µ*. This is true
for all*j=*1*,*2*,...,n*. From above,*ν≥νj* for some*j* between 1 and*n*. Now,
minimality of*ν*implies*ν=νj*. Therefore,*ν*is a minimal member of Ass*R(µ).*
(*⇐*).*ν≥µ*implies that there exists a minimal prime fuzzy ideal*ω*such that
*ν≥ω≥µ*. Therefore, there exists a*νj∈*Ass*R(µ)*such that*ν≥ω≥νj≥µ*
as stated above. Minimality of*ν*in Ass*R(µ)*implies that*ν=ω=νj*which, in
turn, implies that*ν*is a minimal prime fuzzy ideal containing*µ*.

**Note3.9.** Minimal members of Ass*R(µ)*are called the*minimal*or*isolated*
primes of*µ*. The other members of Ass*R(µ)*are called*embedded* primes of*µ*.

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

**Proof.** _{Clearly,}*ω*_{:}*aλ ≥ω*_{. Let} *rβ∈ω*_{:}*aλ*_{. Then,}*rβaλ∈ω*_{. But, by}

hypothesis,*aλ _{∈}_{ν}_{=}√_{ω}*

_{. Therefore,}

_{r}β_{∈}_{ω}_{since}

_{ω}_{is}

_{ν}_{-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=µ*1*∧µ*2*∧···∧µn,* *withνi=√µifori=*1*,...,n,*

*µ=ξ*1*∧ξ*2*∧···∧ξn,* *withνi=*

*ξifori=*1*,...,n.* (3.1)

**Proof.** _{For}*n=*_{1, Ass}*R(µ)= {ν*_{1}*}*_{. So,}*µ*_{1}*=ξ*_{1}_{. For}*n >*_{1, let}*µi*_{be a minimal}

member of Ass*R(µ)*. Then, there exists a fuzzy point*rλ _{∈}n*

*j=1, j*≠*iνj\νi*.
(Oth-erwise,*νi≥nj=1, j*≠*iνj*which, in turn, will imply that*νi> νj*for some*j*,
con-tradicting the minimality of*νi*.) Now, for* >*0,*rλ− _{∈}_{νi}*

_{and}

_{r}λ−_{∈}n*j=*1*, j*≠*iνj*.
So, we can assume without loss of generality that*rλ*_{is strictly in}*n*

*j=1, j*≠*iνj\νi*.
Thus, there exists a positive integer*mj*such that*(rmj _{)}λ_{∈}_{µj}*

_{for every}

_{j}_{≠}

_{i}_{.}Set

*t≥*max

*{mj*:

*j=*1

*,*2

*,...,n, j*≠

*i}*. Then,

*(rt*

_{)}λ_{∈}_{µj}_{and}

_{(r}t_{)}λ_{∈}_{νi}_{. }There-fore,

*µ*:

*(rt*

_{)}λ_{ =}_{(µj)}_{:}

_{(r}t_{)}λ_{ =}_{µi}_{:}

_{(r}t_{)}λ_{}_{since}

_{µj}_{:}

_{(r}t_{)}λ_{ =}_{R}_{for}

_{j}_{≠}

_{i}_{.}Now,

*µi*:

*(rt*

_{)}λ_{ =}_{µi}_{by}

_{Lemma 3.10}

_{since}

_{µi}_{is assumed to be}

_{νi}_{-primary.}Similarly,

*ξi=µi*:

*(rt*

_{)}λ_{}_{for a large enough value for}

_{t}_{. Thus, we have}

_{µi}_{=}_{ξi}_{.}

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

**Definition _{4.1}.**

_{Let}

*µ*

_{be a fuzzy ideal of}

*R*

_{. It is said to be}

_{irreducible}_{if}

*µ*≠

*R*and whenever

*µ=µ*1

*∧µ*2, where

*µ*1and

*µ*2are fuzzy ideals of

*R, then*

*µ=µ*1or*µ=µ*2.

**Proposition4.2.** *Letµ* *be an irreducible fuzzy ideal ofR. Then,*Im*(µ)=*
*{*1*,t}for some*0*≤t <*1*, and the top cutµ*1_{is irreducible.}

**Proof.** _{Suppose that}*µ(z)=t*_{1}*< t*_{2}*=µ(y)*≠_{1 for some}*z,y∈R*_{. Deﬁne}

*µ*1*(x)=*

1*,* *x∈µ*1_{,}

*µ(x), x∈µt*2*\µ*1*,*

*t*2 otherwise*,*

*µ*2*(x)=*

1*,* *x∈µt*2*,*

*µ(x),* otherwise.

(4.1)

Firstly, we note that*µ*1*,µ*2*≥µ*and*µ*1≠*µ*,*µ*2≠*µ*. Secondly,

*µ*1*∧µ*2*(x)=*

1, *x∈µ*1_{,}

*µ(x),* otherwise
*=µ(x).*

(4.2)

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

Further, suppose that*µ*1_{is reducible. Then, there exist two proper ideals}* _{J}*
1

and*J*2of*R*such that*µ*1*=J*1*∩J*2. For*i=*1*,*2, let

*µi(x)=*

1*,* *x∈Ji,*

We note that*µi≥µ*and*µi*≠*µ*for*i=*1*,*2,

*µ*1*∧µ*2

*(x)=*

1*,* *x∈µ*1_{,}

*µ(x)* otherwise
*=µ(x).*

(4.4)

This is a contradiction to the irreducibility of*µ. Therefore,µ*1_{is irreducible.}

With each irreducible crisp ideal of*R*, there is an associated irreducible fuzzy
ideal of*R*as the following proposition shows.

**Proposition _{4.3}.**

_{Let}I_{be an irreducible ideal of}R_{and define a fuzzy ideal}*µasµ(x)=*1

*ifx∈Iandαifx∈Ifor some*0

*≤α <*1

*. Thenµis irreducible.*

**Proof.** Suppose that there exist fuzzy ideals *µ*_{1}*,µ*_{2}≠ *µ* such that *µ* *=*
*µ*1*∧µ*2. This implies the existence of an*x∈R*such that*µ*1*(x) > µ(x)*.

Simi-larly, there is a*y∈R*such that*µ*2*(y) > µ(y)*. Consider the ideals generated by

*I*and*x*,*(I,x)*, and*I*and*y*,*(I,y)*. Clearly,*I*is contained in*I,x∩I,y*,
and also, *I* is distinct from each of *I,x*and *I,y*. Let *a∈ I,x ∩ I,y*,
then*a=r*1*x*1*+s*1*y=r*2*x*2*+s*2*x*, where*x*1*,x*2*∈I* and*r*1*,r*2*,s*1*,s*2*∈R*. Now,

*µ(r*1*x*1*−r*2*x*2*)=*1*=µ(s*2*x−s*1*y)*. This implies that*µ(r*1*x*1*)=µ(r*2*x*2*)*and

*µ(s*2*x)=µ(s*1*y)*. Also,*µ*1*(s*2*x)≥µ*1*(x) > µ(x)≥α*and *µ*2*(s*2*x)=µ*2*(s*2*x−*

*s*1*y+s*1*y)≥µ*2*(s*2*x−s*1*y)∧µ*2*(s*1*y)=µ*2*(s*1*y)≥µ*2*(y) > µ(y)≥α*.

There-fore,*(µ*1*∧µ*2*)(s*2*x) > α*which implies that*µ(s*2*x) > α*. Hence,*µ(s*2*x)=*1. This

implies that*a∈I*and so*I,y ∩ I,x ⊆I*. Therefore,*I= I,y ∩ I,x*. This
contradicts the irreducibility of*I*.

**Corollary _{4.4}.**

*1*

_{If}µ_{is a fuzzy ideal such that}µ

_{is irreducible and}_{Im}

*(µ)=*

*{1,α}, thenµis irreducible.*

**Note _{4.5}.**

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

**Example4.6.** Let*R* be the ringZ*+*Z. Consider*I=*4Z*+*Z. We claim that*I*
is irreducible. Suppose not. Then, there exist two ideals*J*1 and*J*2such that

*Ji*≠*I;I⊂Ji*for*i=*1,2 where*I=J*1*∩J*2. It is an easy exercise to check that

*J*1and*J*2are of the form*k*1Z*+k*2Zand*m*1Z*+m*2Zfor some positive integers

*k*1,*k*2,*m*1and*m*2, respectively. Then, 4Z*+*Z*=(k*1Z*+k*2Z*)∩(m*1Z*+m*2Z*)=*

*(k*1Z*∩m*1Z*)+(k*2Z*∩m*2Z*), which implies triviallyk*2*=*1*=m*2. Further, 4Z*⊂*

*k*1Z*∩m*1Z. Therefore,*k*1,*m*1divide 4. Each possible value for*k*1and *m*1as

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

**Proof.** _{Since}*µ* _{is irreducible,}*µ* ≠*R*_{; Im}*(µ)= {*_{1}*,α}*_{for some 0}*≤α <*_{1}
and*µ*1_{is irreducible. Now, we need only to show that}* _{µ}*1

_{is primary. To this}

end, consider*r s∈µ*1_{and}* _{s}_{∈}_{µ}*1

_{. Since}

_{R}_{is Noetherian, we can ﬁnd a positive}

integer*n*such that the ascending chain*µ*1_{:}* _{r ⊆}_{µ}*1

_{:}

*2*

_{r}*1*

_{ ⊆}_{µ}_{:}

*3*

_{r}

_{ ⊆ ···}ends with*µ*1_{:}* _{r}n_{ =}_{µ}*1

_{:}

_{r}n+1_{ = ···}_{.}

**Claim.** _{The top cut}*µ*1*=(µ*1*+rnR)∩(µ*1*+Rs)*_{.}

Clearly,*µ*1* _{⊆}_{(µ}*1

*1*

_{+}_{r}n_{R)∩}_{(µ}

_{+}_{Rs)}_{. For the reverse inclusion, consider,}

_{y}_{∈}*(µ*1* _{+r}n_{R)∩}_{(µ}*1

_{+Rs)}_{. Then,}

_{y}_{=}_{s}1*+rns*2*=s*3*+ts*for*s*1*,s*3*∈µ*1,*s*2*∈R*, and

*t∈R*. The element*r y=r s*1*+rn+*1*s*2*=r s*3*+t(r s)∈µ*1, which implies that

*rn+1 _{s}*

2*∈µ*1, which, in turn, implies that*s*2*∈µ*1:*rn+1*. Therefore,*s*2*∈µ*1:

*rn _{}*

_{, which implies that}

_{r}n_{s}_{2}

*1*

_{∈}_{µ}_{. Therefore,}

*1*

_{y}_{∈}_{µ}_{. This proves the claim.}

Since*µ*1_{is irreducible, either}* _{µ}*1

*1*

_{=}_{µ}

_{+}_{r}n_{R}_{or}

*1*

_{µ}*1*

_{=}_{µ}

_{+}_{Rs}_{. But}

*1*

_{s}_{∈}_{(µ}

_{+}*Rs)\µ*1_{. Hence,}* _{µ}*1

*1*

_{=}_{µ}

_{+}_{r}n_{R}_{, implying that}

*1*

_{r}n_{R}_{⊆}_{µ}_{. This completes the}

proof.

**Lemma _{4.8}.**

_{Let}µ_{be a fuzzy ideal of}R_{such that}|Im(µ)| =_{2}

_{. Then,}µ

_{can}*be expressed as a finite intersection of irreducible fuzzy ideals ofR.*

**Proof.** The top cut*µ*1is an intersection of a collection of irreducible ideals
*Ji*,*i=*1*,*2*,...,n*of*R*[13]. Suppose that Im*(µ)= {*1*,t}*, 0*≤t <*1. Deﬁne*νi(x)=*
1 if*x∈Ji*and*t*otherwise. Each*νi*is irreducible and*µ=n _{i}_{=1}νi*.

**Theorem _{4.9}.**

_{Let}µ_{be a proper finite-valued fuzzy ideal of}R_{. Then,}µ_{can}*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,...,t*1*}*, where 1*> tn> tn−1>···> t*1.

Let *µ*1 * _{= ∩}m*1

*i=*1*J*1*i*, *µtn* *= ∩mi=*21*Jtni,...,µt*2 *= ∩mi=n*1*Jt*2*i* where *J*11*,...,Jtn*1*,...,*

*Jt*21*,...,Jt*2*mn* are all irreducible ideals of*R*. Deﬁne the fuzzy ideals*ν*’s as
fol-lows:

*ν*1*i(x)=*

1*,* if*x∈J*1*i,*

*tn,* otherwise*,*

*νtni(x)=*

1*,* if*x∈Jtni,*
*tn−1,* otherwise*,*

.. .

*νt*2*i(x)=*

1*,* if*x∈Jt*2*i,*

*t*1*,* otherwise*.*

(4.5)

Then, all the above*ν*1*i,...,νt*2*i,...*are irreducible. Further,

*m*1

*i=1*

*ν*1*i(x)=*

1*,* if*x∈ ∩m*1

*m*2

*i=1*

*νtni(x)=*

1*,* if*x∈ ∩m*2

*i=*1*Jtni=µtn,*
*tn−*1*,* otherwise*,*

.. .

(4.6)

Therefore,*µ=(ν*1*i)∧(νt _{n}i)∧···∧(νt*2

*i).*

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 of*R*can be expressed as a ﬁnite intersection of irreducible fuzzy ideals
of*R*. Now, fromProposition 4.7, we know that an irreducible fuzzy ideal is
primary. Thus, we have the following corollary.

**Corollary _{4.10}.**

_{Let}µ_{be a proper finite-valued fuzzy ideal of}R_{. Then,}µ*has a reduced primary decomposition.*

**5. Krull’s intersection theorem.** In general, it is not true that*∞ _{n}_{=}*1

*µn=*0

for a fuzzy ideal*µ*of*R. 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.

**Lemma _{5.1}.**

_{Let}ν_{be a fuzzy ideal of}R_{. Suppose that}√ν_{is a finitely }*gener-ated fuzzy ideal of*

*R. Then, there exists a natural numbernsuch that(√ν)n*

_{≤ν}_{.}**Proof.** Let*√ν= aλ*_{1}1*,a*_{2}*λ*2*,...,aλ _{k}k*for some positive integer

*k*. Given an arbitrary

*>*0, there exists, for each

*i*between 1 and

*k*,

*ni∈*N such that

*(ani*

*i* *)λi−∈ν*. Set*n=*1*+*

*k*

*i=1(ni−*1*)*. ByProposition 2.2,*(√ν)n={(ar*11*)λ*1*···*

*(ark*

*k)λk*:*r*1*+r*2*+···+rk=n}*, where we assume that*λi=*1 when*ri=*0. It is

easy to see that an*rj≥nj*for some 1*≤j≤n*. Therefore,*(arj*

*j* *)λj−∈ν*. As
is arbitrary,*(√ν)n≤ν.*

**Proposition _{5.2}.**

_{Let}µ

_{be a finite-valued proper fuzzy ideal of}R_{and}ν_{a}*prime fuzzy ideal ofR. Then,ν∈*Ass

*R(µ)if and only if there existsaλ*

_{∈}_{FP}

_{(R)}*such thatµ*:

*aλ*

_{ =}_{ν}_{.}**Proof.** (*⇐*). Suppose that there exists a*aλ∈*FP*(R)*such that*µ*:*aλ =ν*.

Then,*µ*:*aλ _{}*

_{is prime, therefore}

_{ν}_{-primary. Then, by}

_{Theorem 3.5}

_{,}

_{ν}_{=}_{νi}_{for}some 1

*≤i≤n*such that

*νi=√µi*where

*µ=µ*1

*∧µ*2

*∧ ··· ∧µn*is a reduced

primary decomposition, that is,*ν∈*Ass*R(µ)*.

(*⇒*). Suppose that*ν∈*Ass*R(µ)*where*µ=µ*1*∧µ*2*∧ ··· ∧µn*,*νi=√µi*, is a

reduced primary decomposition. Consider*ωj=ni=1,i*≠*jµi*. This implies that
*µ≤ωj≤µj* since*µ=ni=*1*µi* is a reduced primary decomposition. Now, by
Proposition 2.4,*νj*is ﬁnitely generated. So,Lemma 5.1implies that there exists
a positive integer*t*such that*νt*

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

Choose *aλ* * _{∈}_{ν}u−*1

*j* *ωj\µ*. Then, *aλ* *∈ωj\µ*. Now, *µi* :*aλ =R* for all*i=*
1*,*2*,...,n*,*i*≠*j*and*µj*:*aλ _{}*

_{is}

_{νj}_{-primary by}

_{Proposition 3.3}

_{. Thus,}

_{µ}_{:}

_{a}λ_{ =}_{n}

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

*νu*

*jωj≤µ. Hence,νj≤µ*:*aλ ≤νj*since*µ*:*aλ =µj*:*aλ ≤*

*µj*:*aλ =νj*.
This completes the proof.

**Theorem5.3.** *Letµbe a finite-valued fuzzy ideal of* *R, and setν=∞ _{n}_{=}*

_{1}

*µn.*

*Thenν=µν.*

**Proof.** _{The case when}*µ=R*_{is clear. Now, suppose that}*µ*≠*R*_{. Then,}*µν*≠
*R*since*µν≤µ. Thus,µν*has a primary decomposition byCorollary 4.10. Let
*µν* *=µ*1*∧µ*2*∧ ··· ∧µn* with *νi* *=√µi*, *i=*1*,*2*,...,n* be a reduced primary

decomposition of*µν*. We need only to show that *ν* *≤µν* since the reverse
inequality is always true. Suppose that there is a natural number*i*, 1*≤i≤n*,
such that*ν≤µi*. Then, there exists a fuzzy point*aλ _{∈}_{ν}_{\µi}*

_{. Now,}

_{a}λ_{µ}_{≤}_{µν}_{=}_{n}

*i=*1*µi≤µi*and *µi* is*νi*-primary. Therefore,*µ≤√µi=νi* since*aλ∈µi*, and

*νi*is ﬁnitely generated byProposition 2.4. Now, byLemma 5.1, there exists a
positive integer *t* such that*νt*

*i* *≤µi*, thus *ν=*

*∞*

*n=1µn≤µt* *≤νit* *≤µi*. This
contradicts our assumption that*ν≤µi*. Therefore,*ν≤µi*for all*i=*1*,*2*,...,n*,
and hence,*ν≤µν*. This completes the proof.

**Lemma _{5.4}**

_{(Nakayama’s lemma)}

**.**

_{Let}µ

_{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.*

**Note _{5.5}.**

_{By the trivial fuzzy submodule of}

*M*

_{, we mean, for all}

*x∈M*

_{,}

*µ(x)=*1 if

*x=*0 and 0 if

*x*≠0.

**Proof.** _{Since}*µ=νµ, suppµ=(suppν)(suppµ). SetI=*_{supp}*ν*_{be the crisp}

ideal in*R*. Let Mbe any maximal ideal of *R*. Construct *ω*on *R* as follows:
*ω(x)=*1 if*x∈*M and 0 otherwise. Note that*ω* is a maximal fuzzy ideal
of*R. Now,ν* is contained in the fuzzy Jacobson radical of*R* which is itself
contained in*ω*. Therefore,*ν≤ω*, implying*I=*supp*ν⊆*supp*ω=*M. Thus,*I*
is contained in the crisp Jacobson radical of*R*. By Nakayama’s lemma for the
crisp case, supp*µ=*0. This completes the proof.

**Theorem _{5.6}**

_{(Krull’s intersection theorem)}

**.**

_{Let}µ

_{be a finitely generated}*fuzzy ideal ofR*

*such thatµ*

*is contained in the fuzzy Jacobson radical ofR.*

*Then∞*1

_{n}_{=}*µn=*0

*.*

**Proof.** Let *ν=* *∞ _{n}_{=}*

_{1}

*µn*. Since

*µ*is ﬁnitely generated,

*µ*is ﬁnite-valued,

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

**Acknowledgment.** This research was supported by the Foundation for

**References**

[1] T. W. Hungerford,*Algebra*, Graduate Texts in Mathematics, vol. 73,
Springer-Verlag, New York, 1980.

[2] R. Kumar,*Fuzzy irreducible ideals in rings*, Fuzzy Sets and Systems**42**(1991),
no. 3, 369–379.

[3] P. Lubczonok,*Fuzzy vector spaces*, Fuzzy Sets and Systems**38** (1990), no. 3,
329–343.

[4] B. B. Makamba and V. Murali,*On prime fuzzy submodules and radicals*, J. Fuzzy
Math.**8**(2000), no. 4, 831–843.

[5] ,*Primary decompositions of fuzzy submodules*, J. Fuzzy Math.**8**(2000),
no. 4, 817–829.

[6] ,*On an algebra of fuzzy points on modules*, J. Fuzzy Math.**9**(2001), no. 4,
799–806.

[7] D. S. Malik and J. N. Mordeson,*Fuzzy prime ideals of a ring*, Fuzzy Sets and
Systems**37**(1990), no. 1, 93–98.

[8] ,*Fuzzy primary representations of fuzzy ideals*, Inform. Sci. **55**(1991),
no. 1-3, 151–165.

[9] ,*Radicals of fuzzy ideals*, Inform. Sci.**65**(1992), no. 3, 239–252.

[10] ,*R-primaryL-representations ofL-ideals*, Inform. Sci.**88**(1996), no. 1-4,
227–246.

[11] J. N. Mordeson and D. S. Malik,*Fuzzy Commutative Algebra*, World Scientiﬁc
Publishing, New Jersey, 1998.

[12] R. Sekaran,*Fuzzy Ideals in Commutative Rings*, Master’s thesis, Rhodes
Univer-sity, South Africa, 1995.

[13] R. Y. Sharp,*Steps in Commutative Algebra*, London Mathematical Society Student
Texts, vol. 19, Cambridge University Press, Cambridge, 1990.

[14] U. M. Swamy and K. L. N. Swamy,*Fuzzy prime ideals of rings*, J. Math. Anal. Appl.
**134**(1988), no. 1, 94–103.

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

*E-mail address*:v.murali@ru.ac.za

B. B. Makamba: Department of Mathematics, University of Fort Hare, Alice 5700, South Africa