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Primary fuzzy ideals and radical of fuzzy ideals
Fuzzy Sets and Systems 56 (1993) 97-101 North-Holland
97
Primary fuzzy ideals and radical of fuzzy ideals T.K. Mukherjee Department of Mathematics, Jadavpur University, Calcutta 700 032, India
M.K. Sen Department of Pure Mathematics, University of Calcutta, 35 Ballygunge Circular Road, Calcutta 700 019, India Received February 1988 Revised April 1992
Abstract: The aim of this paper is to introduce primary fuzzy ideals and radical of fuzzy ideals in rings. Keywords: Fuzzy ideal; w.c. prime fuzzy ideal; primary fuzzy ideal.
1. Introduction In his classical paper [9], Zadeh introduced the notion of a fuzzy subset A of a set X as a function from X into [0, 1]. Rosenfeld [7] used this concept to study the elementary theory of groupoids and groups. Kuroki [2, 3, 4] studied fuzzy ideals and bi-ideals in semigroups. Wang-Jin Liu [8] introduced fuzzy ideals in a ring. Following [(8), p. 138] a non-empty fuzzy subset .4 of a ring R is called a fuzzy left (right) ideal of a ring R iff A(x - y) .>/min(.4 (x), A(y)) and
f~(xy) >~A ( y )
(respectively ft(xy) >~_4(x)).
In [5] we investigated some interesting properties of fuzzy ideals, where among other results we characterised regular rings by fuzzy ideals and determined all fuzzy ideals of the ring Z of integers. In [6] we determined completely all prime fuzzy ideals of a ring. In this paper we want to introduce primary fuzzy ideals and radical of fuzzy ideals in rings.
2. Fuzzy primary ideals Throughout the section our hypothesis will restrict to commutative rings.
Definition. A fuzzy ideal A of a ring R is called a primary fuzzy ideal if for a, b ER, f~(ab)>~f~(a) implies A(b n) --'I>A(ab) for some positive integer n. Theorem 1. Let I be an ideal of a ring R. Its characteristic function A1 is a fuzzy primary ideal of R iff I is a primary ideal. Correspondence to: Dr. M.K. Sen, Department of Pure Mathematics, University of Calcutta, 35 Ballygunge Circular Road, Calcutta 700 019, India. 0165-0114/93/$06.00 © 1993--Elsevier Science Publishers B.V. All rights reserved
T.K. Mukherjee, M.K. Sen / Primaryfuzzy ideals and radical of fuzzy ideals
98
Proof. Let I be a primary ideal of R. Suppose for a, b c R, hz(ab) > hi(a). Since hi(x) is either zero or 1, we find from ht(ab) > hi(a) that hi(a) = 0 and hl(ab) = 1. Hence ab e l and a ~ L Since I is primary ([1], p. 81), b" c I for some positive integer n. Hence hz(b ") = 1 = h~(ab), so that h~ is a fuzzy primary ideal of R. Conversely assume that hi is a fuzzy primary ideal. Let ab c I and a ~ L Then h1(ab)--1 and h1(a) = 0. Hence Al(ab)> A/(a). Since At is fuzzy primary, there exists a positive integer n such that )tl(b n) t> M(ab). Hence h/(b ~) = 1. This shows that b " c L Consequently I is primary. Theorem 2. Let A be a fuzzy ideal of a ring R. Then A is a fuzzy primary ideal iff A' = {x e R: A ( x ) >t t} is a primary ideal of R for all t c [0, A (0)]. Proof. For any fuzzy ideal A, it is known [6] that A' is an ideal of R. Suppose now A is a fuzzy primary ideal of R. Let ab e A t and a ~t A t. Then A(ab) >>-t and A(a) < t. Hence A(ab) > A ( a ) . Since A is fuzzy primary, there exists a positive integer n such that A ( b n) >~A(ab)>! t. Then b n e A t, so that A t is a primary ideal. Conversely, assume that At is a primary ideal of R for all t c [0, A(0)]. Suppose A (ab) > A (a) where a, b c R . Let t = A ( a b ) . Then t e [ 0 , A(0)], a b c A ' and t > A ( a ) implies that a ¢ A t. From the assumption, there exists a positive integer n such that b ~ c A t. Hence A(b n) >~t = A ( a b ) . This proves that A is a primary fuzzy ideal.
Let us recall [6] the following definition. Definition. A fuzzy ideal A of a ring R is said to be a weakly completely prime fuzzy ideal (abbreviation: w.c. fuzzy prime) if A : R --> [0, 1] is not a constant map and if for x, y c R
A ( x y ) = max(A(x), A(y)). Let A be a w.c. prime fuzzy ideal of R. Suppose a, b are two elements of R such that A(ab) > A ( a ) . But A ( a b ) = max(A(a), A(b)). Hence A(ab) = A ( b ) . Hence A is a primary fuzzy ideal. Now the following example shows that the converse is not true. Example. Let Z be the ring of integers. Suppose
Ra=2Z={2n:neZ},
R2--4Z
and
Ro=Z\R1.
Define a fuzzy subset A on Z by A(m) =
1 1
i f m ER2, ifm cRI\R2,
0
i f m eR0.
We can show that A is a fuzzy primary ideal but not a fuzzy w.c. prime fuzzy ideal of Z.
3. The radical of a fuzzy ideal Definition. Let A be a fuzzy ideal of a ring R. Then X/A, called the radical of A, is defined by
VA (x) = sup A (xn). n~l
Theorem 3. Let A be a fuzzy ideal of a ring R. Then ~/A is a fuzzy ideal of R.
T.K. Mukherjee, M.K. Sen / Primaryfuzzy ideals and radicalof fuzzy ideals
99
Proof. We have
= nEZ +sup (A(x-y)n)=SUPn
C.;~ ( x - y )
[A(E
(l'lr)Xn(--Y)n--r)]
~ s u p [ ~ n A ( x r y n - r ) ] ~>sup (A(xr - ny n r,)) where 0 ~< rn ~
min(CA (x), CA (y)).
Again
C A (xy) = sup (A(xny")) >i sup (A(x"), A ( y " ) ) = sup(CA (x), CA (y)). n~g
+
n
Hence CA is a fuzzy ideal of R. T h e o r e m 4. Let I be an ideal of a ring R. Then CA~ = )tvt denotes the characteristic function of I and CI
denotes the radical of the ideal L Proof. Let x e R. CAt (x) = sup, eZ+()~l(Xn)) 1 or 0. Suppose CA1 (x) = 1. There exists n e Z + such that h t ( x " ) = l . Hence x " e l , so that x eCI. This implies that h , / ~ ( x ) = l . Again assume that C A 1 (x) =- 0. Then h~(x n) = 0 for all n e Z +. Hence x" ¢ I, for all n E Z +. This implies that x ~ CI. Then h,z1(x) = 0. Thus we find that CAt = A,/1. ~-
T h e o r e m 5. I r A is a fuzzy w.c. prime ideal of R then CA = A.
Proof. The proof follows from A(x 2) = sup(A(x), A(x))
for all x ~R,
when A is a fuzzy w.c. prime ideal of R. Let us now show that to every primary fuzzy ideal there corresponds a specific w.c. prime fuzzy ideal. T h e o r e m 6. If A is a primary fuzzy ideal then C A is a w.c. prime fuzzy ideal.
Proof. It is known from T h e o r e m 3 that C A is a fuzzy ideal of R. Suppose CA is not a w.c. prime fuzzy ideal of R. There exist x, y e R such that
C A (xy) > m a x ( C A (x), C A (y)). Then sup A (x~y ~) >>-sup A (x~), sup A (yn). nEZ
+
n
(1)
n
Let a o = s u p , A(x~). Since 7t(xny ~) is a monotonically ascending bounded sequence, A(xny ~) is
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T.K. Mukherjee, M.K. Sen / Primaryfuzzy ideals and radical of fuZZy ideals
convergent and there exists no such that for n >I no, fl(x~y ") > ao ~.4(x"). Since A is a primary, there exists a positive integer m, such that ,4 (y~m,) >>_71(x"y") for n ~>no. Hence SUpnx] (yn) ~ sup .~ (xnyn). This contradicts (1). Therefore X/,4 is a w.c. prime fuzzy ideal of R. Theorem 7. Let A1, A2 . . . . ,-4r be primary fuzzy Meals of a ring R. I f for each a e R there exists a positive integer no such that
Al(a') = A 2 ( a ' ) . . . . .
,Ar(a t) for all t >t no
then f~ = 4 1 N - 4 2 N • • • fq-4r is a primary fuzzy ideal with ~/ft = X/fti, i = 1, 2 , . . . , r. Proof. Let a, b E R such that A (ab) > ,4 (a). Then mini A i(ab) > mini A i(a). Let min l~i~r
Ai(ab) = Am(ab)
and
min .4i(a) = Ap(a).
l~i~r
Then fi~p(ab ) >1flm(ab ) > fi~p(a). Now Ap is primary fuzzy ideal, so .4p(b") >1.Ap(ab ) >1Ytm(ab ) = fi~(ab ). But Ap(b n+i) ~filp(b n) f o r all i/>0. Hence from the assumption there exists a positive integer t such that
fi, i(b t) >~.A(ab)
for i = 1, 2 . . . . , r.
Then 71(b t) = minifi~i(b t) ~fi,(ab), so that .4 is a primary fuzzy ideal of R. Again .4 c . 4 i implies that X/fi, ~ / A i . Suppose there exist x • R and an i ( l < i < r ) such that ~ / A ( x ) < X / A i ( x ) . Then sup, .4 (x") < sup, 7ti(x"). Hence
(m n ;,(gnU) < s,p But from the assumption, for x e R, there exists no such that Aj(x") = A ( x " )
f o r . / = 1, 2 . . . . . r and for n >no.
Hence sup, ~ i ( x n) < SUpn Ai(x"), a contradiction. Consequently, X/A = x/ii i, i = 1, 2 , . . . , r. Definition. Let .4 a n d / ) be two non-empty fuzzy subsets of a ring R. Then .4 is said to be fuzzy disjoint f r o m / ) i f , 4 ( z ) . / 3 ( z ) = 0 for all z eR. Lemma. Let ft be a fuzzy ideal o f the ring R. Suppose B is a fuzzy subset o f R such that B (xy)>I rain(/3 (x),/~ (y)). I f A is fuzzy disjoint from B there exists a w.c. prime fuzzy ideal P o f R such that .4 ~ P and B is disjoint from P.
Proof. Let c~ = { I : i is a fuzzy ideal of R, ,4 c 7 and i is disjoint from/~}. This is a non-empty collection. q¢ is partially ordered b y / 3 ~ P(x), P(y). Define fuzzy subsets/~1 and 92 by /~l(Z)=/5(xy) =0
B2(z ) = P(xy) =0
ifzE{rx+nx;rERandneZ}, otherwise, i f z ~ {ry + ny; r e R a n d n eZ}, otherwise.
We can show easily that /~1 and/~2 are both fuzzy ideals of R. Consider the fuzzy ideals P + J~i where (~) 7t- B i ) ( x ) : supx=,+.{min(P(u),/'~i(V))} for all x E R.
T.K. Mukherjee, M.K. Sen / Primary fuzzy ideals and radical of fuzzy ideals
101
Now P ~ / 5 +/)i, because (P +/)i)(x) = sup {min(P(u),/)/(v))} X~u+t]
/> min(P(0),/)i(x)) > min(P(0), P(x))
> P(x),
since P(0) t> P(xy) > P(x).
Similarly P ~ P +/)2. Hence P +/)1 and P + / ) z are not disjoint f r o m / ) . So there exist z~ and z2 such that (P +/),)(Zl)/)(z,) ~ 0 and (P +/)2)(Zz)/)(zz) ~ 0. Then (P+/)l)(Zl)#0,
(P+/)2)(z2)~0,
/)(Zl)~0
and
/)(z2)#O.
(2)
(/5 +/)~)(z) ~ 0 shows that SUpz,=,,+,,{min(P(Ul),/)(vl))} ~ 0. Then there exists zl = u~ + v~ such that P(ul) ~ 0, /)(v~)# 0. Hence vl = r~x + nlx for some rl ~ R and n~ E Z. Similarly there exist uz, v2 ~ R such that zz = u2 + v2 where vz = r2y + nzy and/5(uz) ¢:0, /)(v2) # 0. Now, P(Zl " z2)= P((ul + rlx + nlx)(u2 + r2y + n2y)) i> min{P(ulu2), ['(ulrzy), P(n~u, y), P(r,xu2), P(rlr2xy),
P(n2rlxy), P(nlxu2), P(nlr2xy), P(nlnzxy)} /> min{P(ul), P(u2), P(xy)} 0
(from (1) and the fact that P(u~) # O, P(u2) # 0),
/)(z~ z2)/> min{/)(z,),/)(z2)} ~ 0
(from (2)).
Thus we find that P cannot be disjoint from ft.. This contradiction shows that t5 is a w.c. prime fuzzy ideal of R. Notes. (1) One can show that the maximal element/5 of the above lemma is a prime fuzzy ideal.
(2) In the proof of above lemma we did not assume any known results on usual ideals of a ring. The proof is on the basis of fuzzy set theoretic arguments. (3) If we asume some known results of prime ideals of ring, the proof can be simplified. Now using the above lemma we can prove the following: T h e o r e m 8. Let f~ be any fuzzy ideal. Then X/.7t = (-'){/) is a w.c. prime fuzzy ideal of R and A ~ /)}.
References [1] [2] [3] [4] [5] [6] [7] [8] [9]
D.M. Burton, A First Course in Rings and Ideals (Addison-Wesley, Reading, MA, 1970). N. Kuroki, Fuzzy bi-ideals in semigroups, Comment. Math. Univ. St. Paul. 28 (1980) 17-21. N. Kuroki, On fuzzy ideals and fuzzy bi-ideals in semigroups, Fuzzy Sets and Systems 5 (1981) 203-215. N. Kuroki, Fuzzy semiprime ideals in semigroup, Fuzzy Sets and Systems 8 (1982) 71-79. T.K. Mukherjee and M.K. Sen, On fuzzy ideals of a ring, I, Fuzzy Sets and Systems 20 (1987) 1-7. T.K. Mukherjee and M.K. Sen, Prime fuzzy ideals in rings, Fuzzy Sets and Systems 32 (1989) 337-341. A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl. 35 (1971) 512-517. Wang-Jin Liu, Fuzzy invariant subgroups and fuzzy ideals, Fuzzy Sets and Systems 8 (1982) 133-139. L.A. Zadeh, F'uzzy sets, Inform. and Control 8 (1965) 338-353.
97
Primary fuzzy ideals and radical of fuzzy ideals T.K. Mukherjee Department of Mathematics, Jadavpur University, Calcutta 700 032, India
M.K. Sen Department of Pure Mathematics, University of Calcutta, 35 Ballygunge Circular Road, Calcutta 700 019, India Received February 1988 Revised April 1992
Abstract: The aim of this paper is to introduce primary fuzzy ideals and radical of fuzzy ideals in rings. Keywords: Fuzzy ideal; w.c. prime fuzzy ideal; primary fuzzy ideal.
1. Introduction In his classical paper [9], Zadeh introduced the notion of a fuzzy subset A of a set X as a function from X into [0, 1]. Rosenfeld [7] used this concept to study the elementary theory of groupoids and groups. Kuroki [2, 3, 4] studied fuzzy ideals and bi-ideals in semigroups. Wang-Jin Liu [8] introduced fuzzy ideals in a ring. Following [(8), p. 138] a non-empty fuzzy subset .4 of a ring R is called a fuzzy left (right) ideal of a ring R iff A(x - y) .>/min(.4 (x), A(y)) and
f~(xy) >~A ( y )
(respectively ft(xy) >~_4(x)).
In [5] we investigated some interesting properties of fuzzy ideals, where among other results we characterised regular rings by fuzzy ideals and determined all fuzzy ideals of the ring Z of integers. In [6] we determined completely all prime fuzzy ideals of a ring. In this paper we want to introduce primary fuzzy ideals and radical of fuzzy ideals in rings.
2. Fuzzy primary ideals Throughout the section our hypothesis will restrict to commutative rings.
Definition. A fuzzy ideal A of a ring R is called a primary fuzzy ideal if for a, b ER, f~(ab)>~f~(a) implies A(b n) --'I>A(ab) for some positive integer n. Theorem 1. Let I be an ideal of a ring R. Its characteristic function A1 is a fuzzy primary ideal of R iff I is a primary ideal. Correspondence to: Dr. M.K. Sen, Department of Pure Mathematics, University of Calcutta, 35 Ballygunge Circular Road, Calcutta 700 019, India. 0165-0114/93/$06.00 © 1993--Elsevier Science Publishers B.V. All rights reserved
T.K. Mukherjee, M.K. Sen / Primaryfuzzy ideals and radical of fuzzy ideals
98
Proof. Let I be a primary ideal of R. Suppose for a, b c R, hz(ab) > hi(a). Since hi(x) is either zero or 1, we find from ht(ab) > hi(a) that hi(a) = 0 and hl(ab) = 1. Hence ab e l and a ~ L Since I is primary ([1], p. 81), b" c I for some positive integer n. Hence hz(b ") = 1 = h~(ab), so that h~ is a fuzzy primary ideal of R. Conversely assume that hi is a fuzzy primary ideal. Let ab c I and a ~ L Then h1(ab)--1 and h1(a) = 0. Hence Al(ab)> A/(a). Since At is fuzzy primary, there exists a positive integer n such that )tl(b n) t> M(ab). Hence h/(b ~) = 1. This shows that b " c L Consequently I is primary. Theorem 2. Let A be a fuzzy ideal of a ring R. Then A is a fuzzy primary ideal iff A' = {x e R: A ( x ) >t t} is a primary ideal of R for all t c [0, A (0)]. Proof. For any fuzzy ideal A, it is known [6] that A' is an ideal of R. Suppose now A is a fuzzy primary ideal of R. Let ab e A t and a ~t A t. Then A(ab) >>-t and A(a) < t. Hence A(ab) > A ( a ) . Since A is fuzzy primary, there exists a positive integer n such that A ( b n) >~A(ab)>! t. Then b n e A t, so that A t is a primary ideal. Conversely, assume that At is a primary ideal of R for all t c [0, A(0)]. Suppose A (ab) > A (a) where a, b c R . Let t = A ( a b ) . Then t e [ 0 , A(0)], a b c A ' and t > A ( a ) implies that a ¢ A t. From the assumption, there exists a positive integer n such that b ~ c A t. Hence A(b n) >~t = A ( a b ) . This proves that A is a primary fuzzy ideal.
Let us recall [6] the following definition. Definition. A fuzzy ideal A of a ring R is said to be a weakly completely prime fuzzy ideal (abbreviation: w.c. fuzzy prime) if A : R --> [0, 1] is not a constant map and if for x, y c R
A ( x y ) = max(A(x), A(y)). Let A be a w.c. prime fuzzy ideal of R. Suppose a, b are two elements of R such that A(ab) > A ( a ) . But A ( a b ) = max(A(a), A(b)). Hence A(ab) = A ( b ) . Hence A is a primary fuzzy ideal. Now the following example shows that the converse is not true. Example. Let Z be the ring of integers. Suppose
Ra=2Z={2n:neZ},
R2--4Z
and
Ro=Z\R1.
Define a fuzzy subset A on Z by A(m) =
1 1
i f m ER2, ifm cRI\R2,
0
i f m eR0.
We can show that A is a fuzzy primary ideal but not a fuzzy w.c. prime fuzzy ideal of Z.
3. The radical of a fuzzy ideal Definition. Let A be a fuzzy ideal of a ring R. Then X/A, called the radical of A, is defined by
VA (x) = sup A (xn). n~l
Theorem 3. Let A be a fuzzy ideal of a ring R. Then ~/A is a fuzzy ideal of R.
T.K. Mukherjee, M.K. Sen / Primaryfuzzy ideals and radicalof fuzzy ideals
99
Proof. We have
= nEZ +sup (A(x-y)n)=SUPn
C.;~ ( x - y )
[A(E
(l'lr)Xn(--Y)n--r)]
~ s u p [ ~ n A ( x r y n - r ) ] ~>sup (A(xr - ny n r,)) where 0 ~< rn ~
min(CA (x), CA (y)).
Again
C A (xy) = sup (A(xny")) >i sup (A(x"), A ( y " ) ) = sup(CA (x), CA (y)). n~g
+
n
Hence CA is a fuzzy ideal of R. T h e o r e m 4. Let I be an ideal of a ring R. Then CA~ = )tvt denotes the characteristic function of I and CI
denotes the radical of the ideal L Proof. Let x e R. CAt (x) = sup, eZ+()~l(Xn)) 1 or 0. Suppose CA1 (x) = 1. There exists n e Z + such that h t ( x " ) = l . Hence x " e l , so that x eCI. This implies that h , / ~ ( x ) = l . Again assume that C A 1 (x) =- 0. Then h~(x n) = 0 for all n e Z +. Hence x" ¢ I, for all n E Z +. This implies that x ~ CI. Then h,z1(x) = 0. Thus we find that CAt = A,/1. ~-
T h e o r e m 5. I r A is a fuzzy w.c. prime ideal of R then CA = A.
Proof. The proof follows from A(x 2) = sup(A(x), A(x))
for all x ~R,
when A is a fuzzy w.c. prime ideal of R. Let us now show that to every primary fuzzy ideal there corresponds a specific w.c. prime fuzzy ideal. T h e o r e m 6. If A is a primary fuzzy ideal then C A is a w.c. prime fuzzy ideal.
Proof. It is known from T h e o r e m 3 that C A is a fuzzy ideal of R. Suppose CA is not a w.c. prime fuzzy ideal of R. There exist x, y e R such that
C A (xy) > m a x ( C A (x), C A (y)). Then sup A (x~y ~) >>-sup A (x~), sup A (yn). nEZ
+
n
(1)
n
Let a o = s u p , A(x~). Since 7t(xny ~) is a monotonically ascending bounded sequence, A(xny ~) is
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T.K. Mukherjee, M.K. Sen / Primaryfuzzy ideals and radical of fuZZy ideals
convergent and there exists no such that for n >I no, fl(x~y ") > ao ~.4(x"). Since A is a primary, there exists a positive integer m, such that ,4 (y~m,) >>_71(x"y") for n ~>no. Hence SUpnx] (yn) ~ sup .~ (xnyn). This contradicts (1). Therefore X/,4 is a w.c. prime fuzzy ideal of R. Theorem 7. Let A1, A2 . . . . ,-4r be primary fuzzy Meals of a ring R. I f for each a e R there exists a positive integer no such that
Al(a') = A 2 ( a ' ) . . . . .
,Ar(a t) for all t >t no
then f~ = 4 1 N - 4 2 N • • • fq-4r is a primary fuzzy ideal with ~/ft = X/fti, i = 1, 2 , . . . , r. Proof. Let a, b E R such that A (ab) > ,4 (a). Then mini A i(ab) > mini A i(a). Let min l~i~r
Ai(ab) = Am(ab)
and
min .4i(a) = Ap(a).
l~i~r
Then fi~p(ab ) >1flm(ab ) > fi~p(a). Now Ap is primary fuzzy ideal, so .4p(b") >1.Ap(ab ) >1Ytm(ab ) = fi~(ab ). But Ap(b n+i) ~filp(b n) f o r all i/>0. Hence from the assumption there exists a positive integer t such that
fi, i(b t) >~.A(ab)
for i = 1, 2 . . . . , r.
Then 71(b t) = minifi~i(b t) ~fi,(ab), so that .4 is a primary fuzzy ideal of R. Again .4 c . 4 i implies that X/fi, ~ / A i . Suppose there exist x • R and an i ( l < i < r ) such that ~ / A ( x ) < X / A i ( x ) . Then sup, .4 (x") < sup, 7ti(x"). Hence
(m n ;,(gnU) < s,p But from the assumption, for x e R, there exists no such that Aj(x") = A ( x " )
f o r . / = 1, 2 . . . . . r and for n >no.
Hence sup, ~ i ( x n) < SUpn Ai(x"), a contradiction. Consequently, X/A = x/ii i, i = 1, 2 , . . . , r. Definition. Let .4 a n d / ) be two non-empty fuzzy subsets of a ring R. Then .4 is said to be fuzzy disjoint f r o m / ) i f , 4 ( z ) . / 3 ( z ) = 0 for all z eR. Lemma. Let ft be a fuzzy ideal o f the ring R. Suppose B is a fuzzy subset o f R such that B (xy)>I rain(/3 (x),/~ (y)). I f A is fuzzy disjoint from B there exists a w.c. prime fuzzy ideal P o f R such that .4 ~ P and B is disjoint from P.
Proof. Let c~ = { I : i is a fuzzy ideal of R, ,4 c 7 and i is disjoint from/~}. This is a non-empty collection. q¢ is partially ordered b y / 3 ~ P(x), P(y). Define fuzzy subsets/~1 and 92 by /~l(Z)=/5(xy) =0
B2(z ) = P(xy) =0
ifzE{rx+nx;rERandneZ}, otherwise, i f z ~ {ry + ny; r e R a n d n eZ}, otherwise.
We can show easily that /~1 and/~2 are both fuzzy ideals of R. Consider the fuzzy ideals P + J~i where (~) 7t- B i ) ( x ) : supx=,+.{min(P(u),/'~i(V))} for all x E R.
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101
Now P ~ / 5 +/)i, because (P +/)i)(x) = sup {min(P(u),/)/(v))} X~u+t]
/> min(P(0),/)i(x)) > min(P(0), P(x))
> P(x),
since P(0) t> P(xy) > P(x).
Similarly P ~ P +/)2. Hence P +/)1 and P + / ) z are not disjoint f r o m / ) . So there exist z~ and z2 such that (P +/),)(Zl)/)(z,) ~ 0 and (P +/)2)(Zz)/)(zz) ~ 0. Then (P+/)l)(Zl)#0,
(P+/)2)(z2)~0,
/)(Zl)~0
and
/)(z2)#O.
(2)
(/5 +/)~)(z) ~ 0 shows that SUpz,=,,+,,{min(P(Ul),/)(vl))} ~ 0. Then there exists zl = u~ + v~ such that P(ul) ~ 0, /)(v~)# 0. Hence vl = r~x + nlx for some rl ~ R and n~ E Z. Similarly there exist uz, v2 ~ R such that zz = u2 + v2 where vz = r2y + nzy and/5(uz) ¢:0, /)(v2) # 0. Now, P(Zl " z2)= P((ul + rlx + nlx)(u2 + r2y + n2y)) i> min{P(ulu2), ['(ulrzy), P(n~u, y), P(r,xu2), P(rlr2xy),
P(n2rlxy), P(nlxu2), P(nlr2xy), P(nlnzxy)} /> min{P(ul), P(u2), P(xy)} 0
(from (1) and the fact that P(u~) # O, P(u2) # 0),
/)(z~ z2)/> min{/)(z,),/)(z2)} ~ 0
(from (2)).
Thus we find that P cannot be disjoint from ft.. This contradiction shows that t5 is a w.c. prime fuzzy ideal of R. Notes. (1) One can show that the maximal element/5 of the above lemma is a prime fuzzy ideal.
(2) In the proof of above lemma we did not assume any known results on usual ideals of a ring. The proof is on the basis of fuzzy set theoretic arguments. (3) If we asume some known results of prime ideals of ring, the proof can be simplified. Now using the above lemma we can prove the following: T h e o r e m 8. Let f~ be any fuzzy ideal. Then X/.7t = (-'){/) is a w.c. prime fuzzy ideal of R and A ~ /)}.
References [1] [2] [3] [4] [5] [6] [7] [8] [9]
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