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Fuzzy prime spectrum of a ring
Fuzzy Sets and Systems 46 (1992) 147-154 North-Holland
147
Fuzzy prime spectrum of a ring Rajesh Kumar Department of Mathematics', P.G.D.A. V. College (E), University of Delhi, Nehru Nagar, Ring Road, New Delhi 110065, India Received January 1990 Revised April 1990
Abstract: The set of all fuzzy prime ideals of a commutative ring with identity is topologized and the resulting space is studied.
Keywords: Fuzzy semiprime ideal; fuzzy prime ideal; fuzzy maximal ideal; fuzzy nil radical of a fuzzy ideal; f-invariant fuzzy subset; fuzzy point.
1. Introduction
Since Rosenfeld [14] introduced fuzzy sets in the realm of group theory, many researchers are engaged in fuzzifying various concepts/results of algebra. However, not all the results of group theory and ring theory can be extended to the fuzzy setting (see [3 and 4]). In [11], Liu introduced and studied fuzzy subrings and fuzzy ideals. Subsequently, among others, Yue [16], Mukherjee and Sen [13] and Swamy and Swamy [15] applied some basic concepts from ring theory and developed the theory of fuzzy ideals. These studies were further carried out in [4, 6, 7, 8]. Throughout the paper, R and R' denote commutative rings with identity. In Section 2, some definitions and results to be used in the sequel are given. In Section 3, a topology is defined on the set of all fuzzy prime ideals of a ring and the resulting space, denoted by F Spec R, is shown to be To. A base for the topology of F S p e c R is also obtained. For o~~ [0, 1[, the subspace A = {/~ 6 F S p e c R J lm/u = {1, ol}} is shown to be compact. In addition, the subspace A turns out to be T1 iff every element of A is a fuzzy maximal ideal of R. Moreover, if
the ring R is Boolean, then the subspace A is shown to be compact, Hausdorff and zero dimensional. In Section 4, for a homomorphism f from R onto R', it is shown that F Spec R' is homeomorphic to the subset of F S p e c R consisting of f-invariant elements of F Spec R. In Section 5, irreducible and connectedness properties of F Spec R are discussed. It turns out that F S p e c R is irreducible iff the fuzzy nil radical of R is a fuzzy prime ideal of R; and F S p e c R is disconnected iff R has a nontrivial idempotent element.
2. Preliminaries
In this section, some definitions, results and notations which will be needed later on are presented. 2.1. Definition [17]. Let /~ be any fuzzy subset of a set S and let t c [0, 1]. The set
#t = {x e S J # ( x ) > - t } is called a level subset of/~. Let /~ be any fuzzy ideal of R and let t c [0,/~(0)]. Then the level subset ~, is an ideal of R and is called a level ideal of/~. 2.2. T h e o r e m [16]. A fuzzy subset I~ of R is a fuzzy ideal of R iff each level subset/~,, t ~Im/~, is an ideal of R. [] Throughout the paper, by a prime ideal P of R, we mean that P ~ R and for all a, b c R, the condition ab ~ P implies that either a c P or b~P. 2.3. Definition [13]. A nonconstant fuzzy ideal of R is called fuzzy prime if for any two fuzzy ideals o and 0 of R, the condition o 0 ~_/~ implies that either cr c__/~ or 0 c_ t~.
0165-0114/92/$05.00 ~ 1992--Elsevier Science Publishers B.V. All rights reserved
148
R. Kumar / Fuzzy prime spectrum of a ring
2.4. Theorem [15]. A fuzzy ideal ~t of R is fuzzy prime iff Im/* = {1, t}. t • [0, 1[, and the
ideal Iu = {x • R I/~(x) = 1}
is prime.
Then V(o) = X and V(O) = 0, so that 0, X • T. Next, let 0~ and 02 be any two fuzzy ideals of R. Then
• v ( o , ) u v(o2) 01C 11~ or
[]
02 C2/1
01n02c_~ ~ t~•V(OinO2) 2.5. Definition [4]. A fuzzy ideal ~ of R is called fuzzy maximal if Im ~ = {1, t}, t • [0, 1[, and the ideal
and • V(O 1n
1u = (x • R I ~(x) = 1)
02) ~
0102~_g
01 n 02C2
because 0 1 0 2 c 0 1 N 0 2
01c_/~ or 02_c/~ since # is fuzzy prime
is maximal. 2.6. Definition [15]. Let a and 0 be any two fuzzy ideals of R. The sum a ~3 0 is defined by
• v ( o , ) u v(o2). Hence
( a ~ O)(x)= sup (min(a(a), O(b))), x--a+b
where x, a, b • R. 2.7. Notations. (i) X = { g i l t is a fuzzy prime ideal of R}. (ii) V ( O ) = { ~ • X ] O c _ / ~ } , where 0 is any fuzzy ideal of R. (iii) X(O) = X ~ V(O), the complement of V(O) in X. (iv) For any fuzzy subset a (subset S) of R, ( a ) ((S)) denotes the fuzzy ideal (ideal) generated by a (S). (v) The symbol '0' designates two different elements; the additive identity of a ring and the integer zero. (vi) The symbol [] marks the end of a proof or the end of a statement for which no proof is required.
v ( o,) u v (
= v ( o, n
and thus X(01) n x(02) = X(Ol n 02). This shows that T is closed under finite intersections. Finally, let { O i ] i • A } be any family of fuzzy ideals of R. It can be easily confirmed that
n { v ( o ; ) l i • A} = V ( ( U {Oi l i • A})). In other words,
U {X(O,)li • A} = X ( ( U {0, l i • a } ) ) . Hence T is closed under arbitrary unions. Consequently, T defines a topology on X. [] The topological space (X, T) is called the fuzzy prime spectrum of R and is denoted by F Spec R or, for convenience, by X.
3. A topology on the set X
3.1. Theorem. Let
3.2. Definition [12]. A fuzzy subset/~ of a set S is called a fuzzy point if /~(x) • ]0, 1] for some x•Sandg(y)=0, for a l l y • S - { x } .
T = {X(0) ] 0 is a fuzzy ideal of R). If/t(x) = fl, the fuzzy point/t is denoted by xt~.
Then the pair (X, T) is a topological space. 3.3. Proposition. If f is a homomorphism from Proof. Consider the fuzzy ideals a and 0 of R defined by a ( x ) = 0 and O(x)= 1, for all x • R.
R onto R', then f(xt~ ) = (f(x))t~ for all x • R and
•1o, 1].
149
R. Kumar / Fuzzy prime spectrum of a ring
Proof. Let y e R'. Then
(f(xtO)(y) =
sup x~(r) r~f I(y)
={Off if r = x
(i.e., if
= Hence f(xl~ ) = (.f(x))t~.
whence x~ ~_/~ for all /~ e X. Thus V(x~) = X, i.e. X ( x ~ ) = ~. (iii) Let P and # be same as in the proof of part (ii). Now, if X ( x t ~ ) = X then V(xt~)=0 which implies xt~ 9~/~ and thus fl > ~t(x) so that x ~ P. Hence x ~t U {P I P is prime ideal of R}.
[]
Recall that a topological space Y is compact iff every covering of Y by basic open sets is reducible to a finite subcovering of Y. 3.4. Theorem. (i) X(xtD n X(ytO = X ( ( x y ) t O, where x, y e R and/3 e ]0, 1]. (ii) X ( x ~ ) = ~ if and only if x is nilpotent, where x e R and/3 e ]0, 1]. (iii) X(xt~ ) = X if and only if x is a unit, where x e R and /3 e ]O, 1]. (iv) The subfamily
Ix R,/3 ]o, 1]} of T is a base for T. (v) Let o~ e [0, 1[ and let a = {/u e X I I m / t = {1, o~}}.
Then the subspace A is compact.
Consequently, x is a unit. (iv) Let X ( O ) e T and let /t e X ( 0 ) . Then O(x) >/~(x), for some x e R. Let O(x) = ft. Then x a ~ / ~ and so # e X ( x ~ ) . Now V ( O ) ~ _ V ( x ~ ) , because if a e V(O) then o(x) >>-O(x) = fl = x~(x) so that xa___a and thus a e V(xtO. Hence X(xt~ ) ~_X(O). Thus ~ e X ( x a ) ~_X(O). (v) Proceeding exactly along the same lines as in part (iv), we can easily show that the family
{X(xt,) n a
[ x e R, fl e ]ol, 1]}
constitutes a base for A. Now, let
{X((xi),) N a [i e a and t e K = ]o¢ 1]} be any covering of A by its basic open sets. Let /3 = sup{t ] t e K}. Then the family
(X((xi)~) n a
] i e A}
also covers A. Now,
a = U {X((xi)~) n a Proof. (i) If/~ e X(xt, ) n X(yt~ ) then xt~ ~/~ and
Yt~ 9~/~- This implies /3 > #(x) and /3 > ~ ( y ) so that/3 > ~t(x) = / t ( y ) = l~(xy), since I, = {x e R I/t(x) = 1} is prime, card Im t~ = 2, and /u(a) =/~(b) for all a, b e R - ~ / ~ and x , y , xyq~l u. But then (xy)t~ ~/~ which means /~eX((xy))t~. Since all implications can be reversed, we have proved part (i). (ii) Let P be any prime ideal of R and let/~ be the characteristic function of P. It follows from Theorem 2.4 that ~ e X. Next, if X(xt~ ) = ~ then V(xt~ ) = X which imples x~ ~_ # and therefore /3 ~
[ i e A}
= ( U {X((xi)tJ) I i e A} n A --
(x - v(u
{(x,)a l i ~ A } ) n A
= A -- ( V ( U {(xi)t3 ] i e a } ) N A). This shows that v ( U ((xi)t~ I i e A}) n A = 0. Next, let P be any prime ideal of R. Consider the fuzzy prime ideal/~ of R defined by /~(x)
J"1
if x e P,
[ c~ otherwise.
Clearly,/~ e A and so ¢ V ( U {(xi)t~ l i e a}). Hence (xj)t~ ~/~ for some j e A, whence /3 > /~(xfl. Consequently, xj ¢ P. Hence there is no prime ideal of R containing the set {x~ ] i e A}. Therefore,
({xi I i
A } ) = e.
R. Kumar / Fuzzy prime spectrum of a ring
150
Let e = r~xl + • • • + rnXn, where e is the unity of R and rl, r2. . . . , rn ~ R. Now V ( U {(xi)t~ l i = 1. . . . .
n}) A A = ~,
because if geV(U{(xi)t~ ]i=1,...,
n}) n a
would hold, then U {(xi)t~ [ i = 1 , . . . ,
n} ~_/~
O(x)
and Im/~ = {1, o:}, which would imply /3 =
u(x,)
for all i = l . . . . . n, so that /~(x,-)=l, for all i = 1. . . . . n, since/3 > ol. But then x~ ~ I v for all i = l . . . . . n; in other words e e l , , which is untenable. Now it is a routine matter to confirm that the family { X ( ( x i ) ~ ) n A l i = 1. . . . .
n}
covers A. Hence A is compact.
(iv) Let {/~} be closed. Then V(t~) n A = {t~}, by part (ii). In order to show that /t is fuzzy maximal, we must show that the ideal Iv = {xcRl~u(x)=l} is maximal. For this, it is sufficient to show that there is no prime ideal of R . p r o p e r l y containing 1~,. Let I~ ~ P , for some prime ideal P of R. Consider the fuzzy ideal 0 of R defined by j"1 if x e P, c~ otherwise.
Then 0 ~ A and /~ is properly contained in 0. This contradicts the fact that V(/~) n A = {~}. Conversely, let/~ be fuzzy maximal. Then the ideal l , = { x e R I / ~ ( x ) = l } is maximal. We claim that V ( / 0 A A = {/t}. Clearly, {/~} ~ V(g) n A. Next, if cr~ V(/~) n A then g ___o and thus Iu ~_1o. This means Iu =1o, since 1u is maximal. Hence g = o, since Im/~ = Im o = { 1, ol}.
[]
Recall that a topological space Y is called To, if for any distinct points x and y of Y, either there exists an open set containing x but not y or else there is an open set containing y but not x. Also recall that a topological space is T~ iff every subset consisting of a single point is closed. 3.5. Theorem. (i) The space X is To. (ii) V(g) = {/~}, the closure o f It in X, where g~X. (iii) 0~{/2} if and only if k t ~ O , where g, O e X . (iv) Let A be same as in Theorem 3.4(iv). I f # e A , then {g} is closed in A iff t~ is f u z z y maximal. (In other words, A is Tt iff every element of A is a f u z z y maximal ideal o f R. ) Proof. (i) Let o, O ~ X , cr4=O. Then either o ~ 0 or 0 ~/~. Let cr ~ 0. Then 0 e X ( o ) . Also o ~ X ( o ) and X(cr) is open. So, X is To. (ii) Clearly {fi} ~_ V(/~), since V(/~) is a closed set containing /~. For the reverse inclusion, consider o ~ {/~}. Then, there exists an open set say X ~ V(O) containing o but not g. Therefore 0~o but O c / t ; and so oqiV(l~). Thus V(g) ~_ {fi} and the equality follows. (iii) 0 e {#} if and only if 0 e V(/~), by part (ii) and this is equivalent to/~ ~_ O.
Therefore V(/~) A A = {/~}. Consequently, {#} is a closed subset of A. [] Recall that a topological space Y is Hausdorff, iff for any two distinct points x and y of Y, there exist two disjoint open sets one containing x and the other containing y. Also recall that a topological space is zero dimensional iff it has a base of closed and open sets. 3.6. Theorem. Suppose R is Boolean, ol e [0, 1[, A= {t~eX [Im~=
{1, o~}},
x, y ~ R, and /3 e ]O, 1]. Then: (i) The set X(xt3 ) n A is both open and closed in A , provided that/3 > ol. (ii) X(xt~ ) U X(yt~ ) = S ( z a ) for some z c R. (iii) The space A is Hausdorff. Proof. (i) Since X ( x a ) is open in X, it follows that X(xt~ ) n A is open in A. We now show that X(xt~ ) N A = V((e -x)t~ ) N A . (This would then imply that X(xt~ ) N A is closed in A.) If g e X ( x t ~ ) N A then f l > / ~ ( x ) and I m g = {1, c~} so that g(x) = 0l. H e n c e / 3 > a~ and x ~ I , = {x e R [/~(x) = 1}.
R. Kumar / Fuzzy prime spectrum of a ring This implies that f l > t r and e - x • l u, since x(e-x)=Oel, and the ideal I~ is prime. Consequently, / z ( e - x ) = 1 so that ( e - x ) t ~ ~_ t~ and thus It • V((e -x))t~ N A. Conversely, if g • V((e - x ) , ) fq A then (e - x)t~ ~_/t and I m / ~ = { 1 , cv} which means that fl~<~(e-x). Hence o : < g ( e - x ) and thus #(e - x) = 1. It follows that e - x • Iu and hence x q~I, so that /~(x) = c~ < ft. This means xt~ ~ / t and thus/~ • x(xt~ ) tq A. Hence X(xt~ ) N A = V((e - x)t~ ) N A. (ii) If tt • X(xtO U X(yt~ ) then x~ q~/~ or y~ q~ t~ (which means f l > / ~ ( x ) or f l > j u ( y ) ) . This implies x ~ l ~ or y ~ I , and thus e - x • l , or e - y • 1,. Consequently,
(e-x)(e-y)
=e-x-
y + xy • 1 , ,
so that x + y - xy ~ I~. Hence /~ e X(zt~ ), where z=x+y-xy. (iii) Let / ~ , o e A , /~4:o. Then # and o are fuzzy prime ideals of R and
4. Fuzzy nil radical and algebraic nature of fuzzy prime ideals under homomorphisms 4.1. Definition [14]. Let S and S' be any two sets and let f be any function from S onto S'. A fuzzy subset 0 of S is called f-invariant if
f ( x ) = f ( y ) ~ O(x) = 0 ( y ) , where x, y • S. If 0 is any f-invariant fuzzy subset of S, then f - ' ( f ( O ) ) = O. 4.2. T h e o r e m [4]. Let f be a homomorphism from R onto R' ; !~ be any f -invariant fuzzy prime ideal of R; and It' be any prime ideal of R '. Then f(It) and f ' ( # ' ) are fuzzy prime ideals of R' and R respectively. [] 4.3. Theorem. Let f be a homomorphism from R onto R', X ' = F Spec R',
X* = {I~ • X I l~ is [-invariant},
Im/a = Im cr = {1, a}.
x'(o')
Since every prime ideal of a Boolean ring is maximal, it follows that Iu and Io are maximal ideals of R. So I, ~ Io, since kt 4: o. Choose x • 1~ a n d x $ I o . Then e - x • I o and e - x $ 1 v . Now, o(x) = / t ( e - x ) = tr and/~(x) = 1 = o(e - x ) . Let t • ] o ~ , l [ . Then ( x t ) ( x ) = t > o c = t r ( x ) , so that x, ¢ o. Hence o e X(x,). Also,
(e - x)t(e - x) = t > o: = t~(e - x), so that Further,
151
(e-x),¢/a.
Hence
I~•X((e-x),).
= x'
- v(o'),
where O' is any fuzzy ideal of R ', and g be a map from X ' to X* defined by g ( u ' ) - - f - ' ( t ~ ' ) , It' • X ' . Then: (i) g is continuous, (ii) g is open, and (iii) g is a homeomorphism of X ' onto X*.
Proof. (i): Let #' • X'. It follows from T h e o r e m 4.2 that f-~(/~') • X. Also f I(~LLe)is f-invariant, since for all a , b • R , if f ( a ) = f ( b ) then /~'(f(a)) = / ~ ' ( f ( b ) ) and thus
X(x,) N X ( ( e - x ) , ) = X ( ( x - xZ)t) by T h e o r e m 3.4(i) = X((0),)
since R is Boolean
=0. Consequently, A is Hausdorff.
(f-t(#'))(a) = (f '(g'))(b). Hence g(l~').=f-l(Iz') e X * . Next, let X(xl~ ) (3 X*, f l • ] O , 1], be any basic open set in X*. Then it is easy to verify
[] n x*) =
3.7. Theorem. If R is Boolean, o: • [0, 1[, and a = {/~ • X ]Im U = {1, a~}},
then the subspace A is compact, Hausdorff, and zero dimensional. Proof. Follows immediately 3.4(v) and 3.6(i), (iii). []
from
Theorems
This shows that the inverse image of any basic open set in X* is open in X'. Hence g is continuous. (ii) Let X'((f(x)t~), x e R and f l e ]0, 1], be any basic open set in X ' . Let o e g ( X ' ( ( f ( x ) ) t O ) . Then e = g ( / ~ ' ) = f - l ( ~ ') for some /~' • X ' such that (f(x))~ ~ l~'. As in part (i), we can show
R. Kumar / Fuzzy prime spectrum of a ring
152 that o is f-invariant. Next,
It follows from Theorem 4.5 that ~/it is a fuzzy semiprime ideal of R. The chain of level ideals of ~/it is given by
g(X'((f(x))~)) = X(x~) ¢3X*, because
\//~,~t[, i~_ \ / / i t t i C--- " " " 1~ \//ittm = R .
It • g(X'((f(x))t~)) ¢:> g-l(it) • X,((f(x))t~)
The following example illustrates the above definition.
and It is f-invariant
4.7. Example. Let p, m • Z+, where p is prime. Define a fuzzy subset It of Z (the ring of integers) by
¢:> f(xt~ ) = (f(x))a ¢fig-'(it) = f ( i t ) ¢:> x e q~f - ' ( f ( i t ) ) = It since It is f-invariant
It(x) = ¢,
It • X(x
) n x*.
Hence the direct image of every basic open set in X ' is open in X*; in consequence, g is open. (iii) In view of parts (i) and (ii), it is sufficient to show that g is one-one and onto. Let It', O' e X ' . Then g(it') = g ( O ' ) implies f-~(it') = f - l ( 0 ' ) which means f ( f - l ( # ' ) ) = f ( f - m ( o ' ) ) and thus I t ' = 0', since f is onto. Finally, let It z X*. Then It is an f-invariant fuzzy prime ideal of R and hence (by Theorem 4.2), f ( i t ) is a fuzzy prime ideal of R'. Further
{tti~ i f x • ( p m ) ' i f x • (pro-i>
(pro-i+1), i=1,2 .....
m,
where t i • ] 0 , 1 [ ( i = l . . . . . m) and t 0 > t m > • • • > tm. It is immediate from Theorem 2.2 that It is a fuzzy ideal of R. The chain of level ideals of It is given by
(p")
c
(pro-l)
C''"
C
(pO) = Z.
Since (pk) = ( p ) , K • Z+, it follows that
(\/it)(x)={tto
ifx • (p),
ifx•Z~(p).
g(f(it)) = f - a ( f ( i t ) ) = #, since It is f-invariant. Hence g is onto.
~
[] 4.8. Theorem [8]. Let 0 be any fuzzy ideal of
4.4. Definition [8]. A fuzzy ideal It of R is said to be fuzzy semiprime if It(X n) = It(X), X • R and
n•Z+. 4.5. Theorem [8]. A fuzzy ideal It of R is fuzzy semiprime iff each It,, t • Im It, is a semiprime ideal of R. (ii) A fuzzy prime ideal is always fuzzy semiprime. [] 4.6. Definition [6]. Let It be any fuzzy ideal of R with Im It = {to, tm. . . . .
tin}
and to > tl > • • • > tin. The fuzzy nil radical of It, symbolized by ~/it, is the fuzzy subset of R defined by to (k/it)(x)= ti
ifx • k/itt0,
ifx•X/it,,-X/it, .... i=1 . . . . . m,
R. Then: (i) X/0 is the smallest fuzzy semiprime ideal of R containing O. (ii) X/0 is the intersection of all fuzzy prime ideals of R containing O. [] 4.9. Theorem. Let 0 be any fuzzy ideal of R.
Then
(i)
V(O)=
V(~/O).
(ii) X(xt~)=X(yl~)
if and only if ~/(xt~) = ~/(y~ ), where x, y e R and [3 • ]0, 1].
Proof. (i): Let # • V(O). Since # is fuzzy prime (and hence fuzzy semiprime), it follows from Theorem 4.8(i) that ~/0 ___#. Hence # • V(V/O), so that V(O)~_ V(V~O). The reverse inclusion is obvious. (ii) If X(xt3)-- X(yt~ ) then V(x~)= V(ylO which implies V((x~))= V((yt~)). This means
where X/it,, denotes the nil radical of the ideal
~{#
It,, = {x
and therefore k/(x~) = ~/(Yt~)"
t¢ I It(x) >1 t,}.
] It e V((x~))} = ~ {it [ i t • V ( ( y t , ) ) }
R. Kumar / Fuzzy prime spectrum of a ring
any two n o n e m p t y basic open sets is nonempty. Hence, X i s irreducible. []
Conversely, let "X/(xt~) = k/(yt~ ). T h e n
#•V(xt~) ~
x~=_~ ¢~ (x~)=_~
¢:> k/(xt~) ~- U
153
by T h e o r e m 4.8(i)
5.3. T h e o r e m . The space X is disconnected iff R
has a nontrivial idempotent element. ¢:> yo~_#
as before
Proof. Let X be disconnected. Then there exist fuzzy ideals o and 0 of R such that
¢:> it • V(yt~). Hence
X(Yt,).
V(xt~ ) = V(ylO, []
so
that
X(xt~ ) =
X = V W ) U V(O), V(o)~=O,
5. Irreducibility and connectedness of the space X Recall that a topological space is irreducible iff the intersection of any two n o n e m p t y basic open sets is nonempty. Recall, too, that a topological space is disconnected iff it can be expressed as the union of two non-empty disjoint closed subsets.
V(O)~O,
V(o)nv(o)=~.
Now, V(o) n V(O) = ~ implies V ( o • 0) = ~ so that (o • O)(x) = 1 for all x • R. Hence sup (min(o(a), O(b))) = 1, e=a+b where
e is the
o(a)=O(b)=l e = a + b . Let
identity of R. This means for some a , b • R such that
I = {x • R ] o(x) = 1}, J={x•R
I O(x)= l},
5.1. Definition. The intersection of all the fuzzy prime ideals of R is called the fuzzy nil radical of R and is denoted by/~*.
P be any prime ideal of R, and # be the characteristic function of P. Then/~ • X. Since
5.2. T h e o r e m . The space X
it follows that o n 0 ~/~. Next, if x • 1 N J then (or fq O)(x) = 1 so that #(x) = 1 and thus x • P. This shows that every element of I n J is nilpotent. Now clearly,
is irreducible iff
It* • X . Proof. Let X be irreducible and let N be the nil radical of R. Then #*(x)=
1 0
ifx•N, ifx•R~N.
x = v ( o ) u v ( o ) = v ( o n o),
R / I N J =1/1 n J ~) J / I NJ. Therefore
e+lnJ=g+lNJ+h Next, let a, b ~_--R and let/3 • ]0, 1]. T h e n ab • N implies ab is nilpotent and thus X((ab)~) = ~, by T h e o r e m 3.4(ii). Therefore X(a/3) n X(b~) = f~, by T h e o r e m 3.4(i), which means X(al, ) = ~ or X(bl~ ) = ~, since X is irreducible. Hence either a or b is nilpotent, by T h e o r e m 3.4(ii), and thus a • N or b • N. Consequently, N is prime, whence it follows from T h e o r e m 2.4 that #* • X. Conversely, suppose that g* e X. Then N is pime. Let a , b • R and let f i e ] 0 , 1 ] . Then X(at~ ) NX(bl~ ) = ~ implies X((ab)~) =~, by T h e o r e m 3.4(i), and thus ab is nilpotent, by T h e o r e m 3.4(ii). Then ab • N and so a • N or b • N , which means a is nilpotent or b is nilpotent. Hence X(at~)=~ or X(bt~ ) = ~ , by T h e o r e m 3.4(ii). This shows that intersection of
+lNJ,
so that g 2 _ g • l n J and hence g 2 _ g is nilpotent. Thus (g2 g)m= 0 for some m • Z+. Consequently, gm =gm+lQ(g) for some polynomial Q(g) in g. Let x = gm(Q(g))m. It is a now simple matter to verify that x 4=0, x ~ e, and
X2~X. Conversely, for any nontrivial idempotent element x of R, it can be easily verified that
x = V(x~) u V((e - x)~), V(xt~)~,
v((e-x)t~)~,
V(xt~ ) n V((e - x)l~ ) = ~, where f l • ] 0 , 1]. This disconnected. []
establishes
that
X
is
154
R. Kumar / Fuzzy prime spectrum of a ring
Acknowledgements The author is thankful to the referees for their valuable comments and suggestions.
References [1] P. Das, Fuzzy vector spaces under triangular norms, Fuzzy Sets and Systems, 25 (1988) 73-85. [2] P.S. Das, Fuzzy groups and level subgroups, J. Math. Anal. Appl. 84 (1981) 264-269. [3] V.N. Dixit, Rajesh Kumar and N. Ajmal, Level subgroups and union of fuzzy subgroups, Fuzzy Sets and Systems 37 (1990) 359-371. [4] V.N. Dixit, Rajesh Kumar and N. Ajmal, Fuzzy ideals and fuzzy prime ideals of a ring, Fuzzy Sets and Systems 44 (1991) 127-138. [5] A.K. Katsaras and D.B. Liu, Fuzzy vector spaces and fuzzy topological vector spaces, J. Math. Anal. Appl. 58 (1977) 135-146. [6] R. Kumar, Fuzzy nil radicals and fuzzy primary ideals, Fuzzy Sets and Systems 43 (1991) 81-93.
[7] R. Kumar, Fuzzy irreducible ideals in rings, Fuzzy Sets and Systems 42 (1991) 369-379. [8] R. Kumar, Fuzzy cosets and some fuzzy radicals, Fuzzy Sets and Systems 46 (1992) to appear. [9] N. Kuroki, On fuzzy ideals and fuzzy bi-ideals in semigroups, Fuzzy Sets and Systems 5 (1981) 203-215. [10] N. Kuroki, Fuzzy semiprime ideals in semigroups, Fuzzy Sets and Systems 8 (1982) 71-79. [11] Wang-Jin Liu, Fuzzy invariant subgroups and fuzzy ideals, Fuzzy Sets and Systems 8 (1982) 133-139. [12] Pu Pao-Ming and Liu Ying-Ming, Fuzzy topology I, J. Math. Anal. Appl. 76 (1980) 571-599. [13] T.K. Mukherjee and M.K. Sen, On fuzzy ideals of a ring I, Fuzzy Sets and Systems 21 (1987) 99-104. [14] A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl. 35 (1971) 512-517. [15] K.L.N. Swamy and U.M. Swamy, Fuzzy prime ideals of rings, J. Math. Anal. Appl. 134 (1988) 99-103. [16] Zhang Yue, Prime L-fuzzy ideals and primary L-fuzzy ideals, Fuzzy Sets and Systems 27 (1988) 345-350. [17] L.A. Zadeh, Fuzzy sets, Inform. and Control 8 (1965) 338-353. [18] H.J. Zimmermann, Fuzzy Set Theory - and Its Applications (Kluwer Nijhoff, Dordrecht-Boston, 1985).
147
Fuzzy prime spectrum of a ring Rajesh Kumar Department of Mathematics', P.G.D.A. V. College (E), University of Delhi, Nehru Nagar, Ring Road, New Delhi 110065, India Received January 1990 Revised April 1990
Abstract: The set of all fuzzy prime ideals of a commutative ring with identity is topologized and the resulting space is studied.
Keywords: Fuzzy semiprime ideal; fuzzy prime ideal; fuzzy maximal ideal; fuzzy nil radical of a fuzzy ideal; f-invariant fuzzy subset; fuzzy point.
1. Introduction
Since Rosenfeld [14] introduced fuzzy sets in the realm of group theory, many researchers are engaged in fuzzifying various concepts/results of algebra. However, not all the results of group theory and ring theory can be extended to the fuzzy setting (see [3 and 4]). In [11], Liu introduced and studied fuzzy subrings and fuzzy ideals. Subsequently, among others, Yue [16], Mukherjee and Sen [13] and Swamy and Swamy [15] applied some basic concepts from ring theory and developed the theory of fuzzy ideals. These studies were further carried out in [4, 6, 7, 8]. Throughout the paper, R and R' denote commutative rings with identity. In Section 2, some definitions and results to be used in the sequel are given. In Section 3, a topology is defined on the set of all fuzzy prime ideals of a ring and the resulting space, denoted by F Spec R, is shown to be To. A base for the topology of F S p e c R is also obtained. For o~~ [0, 1[, the subspace A = {/~ 6 F S p e c R J lm/u = {1, ol}} is shown to be compact. In addition, the subspace A turns out to be T1 iff every element of A is a fuzzy maximal ideal of R. Moreover, if
the ring R is Boolean, then the subspace A is shown to be compact, Hausdorff and zero dimensional. In Section 4, for a homomorphism f from R onto R', it is shown that F Spec R' is homeomorphic to the subset of F S p e c R consisting of f-invariant elements of F Spec R. In Section 5, irreducible and connectedness properties of F Spec R are discussed. It turns out that F S p e c R is irreducible iff the fuzzy nil radical of R is a fuzzy prime ideal of R; and F S p e c R is disconnected iff R has a nontrivial idempotent element.
2. Preliminaries
In this section, some definitions, results and notations which will be needed later on are presented. 2.1. Definition [17]. Let /~ be any fuzzy subset of a set S and let t c [0, 1]. The set
#t = {x e S J # ( x ) > - t } is called a level subset of/~. Let /~ be any fuzzy ideal of R and let t c [0,/~(0)]. Then the level subset ~, is an ideal of R and is called a level ideal of/~. 2.2. T h e o r e m [16]. A fuzzy subset I~ of R is a fuzzy ideal of R iff each level subset/~,, t ~Im/~, is an ideal of R. [] Throughout the paper, by a prime ideal P of R, we mean that P ~ R and for all a, b c R, the condition ab ~ P implies that either a c P or b~P. 2.3. Definition [13]. A nonconstant fuzzy ideal of R is called fuzzy prime if for any two fuzzy ideals o and 0 of R, the condition o 0 ~_/~ implies that either cr c__/~ or 0 c_ t~.
0165-0114/92/$05.00 ~ 1992--Elsevier Science Publishers B.V. All rights reserved
148
R. Kumar / Fuzzy prime spectrum of a ring
2.4. Theorem [15]. A fuzzy ideal ~t of R is fuzzy prime iff Im/* = {1, t}. t • [0, 1[, and the
ideal Iu = {x • R I/~(x) = 1}
is prime.
Then V(o) = X and V(O) = 0, so that 0, X • T. Next, let 0~ and 02 be any two fuzzy ideals of R. Then
• v ( o , ) u v(o2) 01C 11~ or
[]
02 C2/1
01n02c_~ ~ t~•V(OinO2) 2.5. Definition [4]. A fuzzy ideal ~ of R is called fuzzy maximal if Im ~ = {1, t}, t • [0, 1[, and the ideal
and • V(O 1n
1u = (x • R I ~(x) = 1)
02) ~
0102~_g
01 n 02C2
because 0 1 0 2 c 0 1 N 0 2
01c_/~ or 02_c/~ since # is fuzzy prime
is maximal. 2.6. Definition [15]. Let a and 0 be any two fuzzy ideals of R. The sum a ~3 0 is defined by
• v ( o , ) u v(o2). Hence
( a ~ O)(x)= sup (min(a(a), O(b))), x--a+b
where x, a, b • R. 2.7. Notations. (i) X = { g i l t is a fuzzy prime ideal of R}. (ii) V ( O ) = { ~ • X ] O c _ / ~ } , where 0 is any fuzzy ideal of R. (iii) X(O) = X ~ V(O), the complement of V(O) in X. (iv) For any fuzzy subset a (subset S) of R, ( a ) ((S)) denotes the fuzzy ideal (ideal) generated by a (S). (v) The symbol '0' designates two different elements; the additive identity of a ring and the integer zero. (vi) The symbol [] marks the end of a proof or the end of a statement for which no proof is required.
v ( o,) u v (
= v ( o, n
and thus X(01) n x(02) = X(Ol n 02). This shows that T is closed under finite intersections. Finally, let { O i ] i • A } be any family of fuzzy ideals of R. It can be easily confirmed that
n { v ( o ; ) l i • A} = V ( ( U {Oi l i • A})). In other words,
U {X(O,)li • A} = X ( ( U {0, l i • a } ) ) . Hence T is closed under arbitrary unions. Consequently, T defines a topology on X. [] The topological space (X, T) is called the fuzzy prime spectrum of R and is denoted by F Spec R or, for convenience, by X.
3. A topology on the set X
3.1. Theorem. Let
3.2. Definition [12]. A fuzzy subset/~ of a set S is called a fuzzy point if /~(x) • ]0, 1] for some x•Sandg(y)=0, for a l l y • S - { x } .
T = {X(0) ] 0 is a fuzzy ideal of R). If/t(x) = fl, the fuzzy point/t is denoted by xt~.
Then the pair (X, T) is a topological space. 3.3. Proposition. If f is a homomorphism from Proof. Consider the fuzzy ideals a and 0 of R defined by a ( x ) = 0 and O(x)= 1, for all x • R.
R onto R', then f(xt~ ) = (f(x))t~ for all x • R and
•1o, 1].
149
R. Kumar / Fuzzy prime spectrum of a ring
Proof. Let y e R'. Then
(f(xtO)(y) =
sup x~(r) r~f I(y)
={Off if r = x
(i.e., if
= Hence f(xl~ ) = (.f(x))t~.
whence x~ ~_/~ for all /~ e X. Thus V(x~) = X, i.e. X ( x ~ ) = ~. (iii) Let P and # be same as in the proof of part (ii). Now, if X ( x t ~ ) = X then V(xt~)=0 which implies xt~ 9~/~ and thus fl > ~t(x) so that x ~ P. Hence x ~t U {P I P is prime ideal of R}.
[]
Recall that a topological space Y is compact iff every covering of Y by basic open sets is reducible to a finite subcovering of Y. 3.4. Theorem. (i) X(xtD n X(ytO = X ( ( x y ) t O, where x, y e R and/3 e ]0, 1]. (ii) X ( x ~ ) = ~ if and only if x is nilpotent, where x e R and/3 e ]0, 1]. (iii) X(xt~ ) = X if and only if x is a unit, where x e R and /3 e ]O, 1]. (iv) The subfamily
Ix R,/3 ]o, 1]} of T is a base for T. (v) Let o~ e [0, 1[ and let a = {/u e X I I m / t = {1, o~}}.
Then the subspace A is compact.
Consequently, x is a unit. (iv) Let X ( O ) e T and let /t e X ( 0 ) . Then O(x) >/~(x), for some x e R. Let O(x) = ft. Then x a ~ / ~ and so # e X ( x ~ ) . Now V ( O ) ~ _ V ( x ~ ) , because if a e V(O) then o(x) >>-O(x) = fl = x~(x) so that xa___a and thus a e V(xtO. Hence X(xt~ ) ~_X(O). Thus ~ e X ( x a ) ~_X(O). (v) Proceeding exactly along the same lines as in part (iv), we can easily show that the family
{X(xt,) n a
[ x e R, fl e ]ol, 1]}
constitutes a base for A. Now, let
{X((xi),) N a [i e a and t e K = ]o¢ 1]} be any covering of A by its basic open sets. Let /3 = sup{t ] t e K}. Then the family
(X((xi)~) n a
] i e A}
also covers A. Now,
a = U {X((xi)~) n a Proof. (i) If/~ e X(xt, ) n X(yt~ ) then xt~ ~/~ and
Yt~ 9~/~- This implies /3 > #(x) and /3 > ~ ( y ) so that/3 > ~t(x) = / t ( y ) = l~(xy), since I, = {x e R I/t(x) = 1} is prime, card Im t~ = 2, and /u(a) =/~(b) for all a, b e R - ~ / ~ and x , y , xyq~l u. But then (xy)t~ ~/~ which means /~eX((xy))t~. Since all implications can be reversed, we have proved part (i). (ii) Let P be any prime ideal of R and let/~ be the characteristic function of P. It follows from Theorem 2.4 that ~ e X. Next, if X(xt~ ) = ~ then V(xt~ ) = X which imples x~ ~_ # and therefore /3 ~
[ i e A}
= ( U {X((xi)tJ) I i e A} n A --
(x - v(u
{(x,)a l i ~ A } ) n A
= A -- ( V ( U {(xi)t3 ] i e a } ) N A). This shows that v ( U ((xi)t~ I i e A}) n A = 0. Next, let P be any prime ideal of R. Consider the fuzzy prime ideal/~ of R defined by /~(x)
J"1
if x e P,
[ c~ otherwise.
Clearly,/~ e A and so ¢ V ( U {(xi)t~ l i e a}). Hence (xj)t~ ~/~ for some j e A, whence /3 > /~(xfl. Consequently, xj ¢ P. Hence there is no prime ideal of R containing the set {x~ ] i e A}. Therefore,
({xi I i
A } ) = e.
R. Kumar / Fuzzy prime spectrum of a ring
150
Let e = r~xl + • • • + rnXn, where e is the unity of R and rl, r2. . . . , rn ~ R. Now V ( U {(xi)t~ l i = 1. . . . .
n}) A A = ~,
because if geV(U{(xi)t~ ]i=1,...,
n}) n a
would hold, then U {(xi)t~ [ i = 1 , . . . ,
n} ~_/~
O(x)
and Im/~ = {1, o:}, which would imply /3 =
u(x,)
for all i = l . . . . . n, so that /~(x,-)=l, for all i = 1. . . . . n, since/3 > ol. But then x~ ~ I v for all i = l . . . . . n; in other words e e l , , which is untenable. Now it is a routine matter to confirm that the family { X ( ( x i ) ~ ) n A l i = 1. . . . .
n}
covers A. Hence A is compact.
(iv) Let {/~} be closed. Then V(t~) n A = {t~}, by part (ii). In order to show that /t is fuzzy maximal, we must show that the ideal Iv = {xcRl~u(x)=l} is maximal. For this, it is sufficient to show that there is no prime ideal of R . p r o p e r l y containing 1~,. Let I~ ~ P , for some prime ideal P of R. Consider the fuzzy ideal 0 of R defined by j"1 if x e P, c~ otherwise.
Then 0 ~ A and /~ is properly contained in 0. This contradicts the fact that V(/~) n A = {~}. Conversely, let/~ be fuzzy maximal. Then the ideal l , = { x e R I / ~ ( x ) = l } is maximal. We claim that V ( / 0 A A = {/t}. Clearly, {/~} ~ V(g) n A. Next, if cr~ V(/~) n A then g ___o and thus Iu ~_1o. This means Iu =1o, since 1u is maximal. Hence g = o, since Im/~ = Im o = { 1, ol}.
[]
Recall that a topological space Y is called To, if for any distinct points x and y of Y, either there exists an open set containing x but not y or else there is an open set containing y but not x. Also recall that a topological space is T~ iff every subset consisting of a single point is closed. 3.5. Theorem. (i) The space X is To. (ii) V(g) = {/~}, the closure o f It in X, where g~X. (iii) 0~{/2} if and only if k t ~ O , where g, O e X . (iv) Let A be same as in Theorem 3.4(iv). I f # e A , then {g} is closed in A iff t~ is f u z z y maximal. (In other words, A is Tt iff every element of A is a f u z z y maximal ideal o f R. ) Proof. (i) Let o, O ~ X , cr4=O. Then either o ~ 0 or 0 ~/~. Let cr ~ 0. Then 0 e X ( o ) . Also o ~ X ( o ) and X(cr) is open. So, X is To. (ii) Clearly {fi} ~_ V(/~), since V(/~) is a closed set containing /~. For the reverse inclusion, consider o ~ {/~}. Then, there exists an open set say X ~ V(O) containing o but not g. Therefore 0~o but O c / t ; and so oqiV(l~). Thus V(g) ~_ {fi} and the equality follows. (iii) 0 e {#} if and only if 0 e V(/~), by part (ii) and this is equivalent to/~ ~_ O.
Therefore V(/~) A A = {/~}. Consequently, {#} is a closed subset of A. [] Recall that a topological space Y is Hausdorff, iff for any two distinct points x and y of Y, there exist two disjoint open sets one containing x and the other containing y. Also recall that a topological space is zero dimensional iff it has a base of closed and open sets. 3.6. Theorem. Suppose R is Boolean, ol e [0, 1[, A= {t~eX [Im~=
{1, o~}},
x, y ~ R, and /3 e ]O, 1]. Then: (i) The set X(xt3 ) n A is both open and closed in A , provided that/3 > ol. (ii) X(xt~ ) U X(yt~ ) = S ( z a ) for some z c R. (iii) The space A is Hausdorff. Proof. (i) Since X ( x a ) is open in X, it follows that X(xt~ ) n A is open in A. We now show that X(xt~ ) N A = V((e -x)t~ ) N A . (This would then imply that X(xt~ ) N A is closed in A.) If g e X ( x t ~ ) N A then f l > / ~ ( x ) and I m g = {1, c~} so that g(x) = 0l. H e n c e / 3 > a~ and x ~ I , = {x e R [/~(x) = 1}.
R. Kumar / Fuzzy prime spectrum of a ring This implies that f l > t r and e - x • l u, since x(e-x)=Oel, and the ideal I~ is prime. Consequently, / z ( e - x ) = 1 so that ( e - x ) t ~ ~_ t~ and thus It • V((e -x))t~ N A. Conversely, if g • V((e - x ) , ) fq A then (e - x)t~ ~_/t and I m / ~ = { 1 , cv} which means that fl~<~(e-x). Hence o : < g ( e - x ) and thus #(e - x) = 1. It follows that e - x • Iu and hence x q~I, so that /~(x) = c~ < ft. This means xt~ ~ / t and thus/~ • x(xt~ ) tq A. Hence X(xt~ ) N A = V((e - x)t~ ) N A. (ii) If tt • X(xtO U X(yt~ ) then x~ q~/~ or y~ q~ t~ (which means f l > / ~ ( x ) or f l > j u ( y ) ) . This implies x ~ l ~ or y ~ I , and thus e - x • l , or e - y • 1,. Consequently,
(e-x)(e-y)
=e-x-
y + xy • 1 , ,
so that x + y - xy ~ I~. Hence /~ e X(zt~ ), where z=x+y-xy. (iii) Let / ~ , o e A , /~4:o. Then # and o are fuzzy prime ideals of R and
4. Fuzzy nil radical and algebraic nature of fuzzy prime ideals under homomorphisms 4.1. Definition [14]. Let S and S' be any two sets and let f be any function from S onto S'. A fuzzy subset 0 of S is called f-invariant if
f ( x ) = f ( y ) ~ O(x) = 0 ( y ) , where x, y • S. If 0 is any f-invariant fuzzy subset of S, then f - ' ( f ( O ) ) = O. 4.2. T h e o r e m [4]. Let f be a homomorphism from R onto R' ; !~ be any f -invariant fuzzy prime ideal of R; and It' be any prime ideal of R '. Then f(It) and f ' ( # ' ) are fuzzy prime ideals of R' and R respectively. [] 4.3. Theorem. Let f be a homomorphism from R onto R', X ' = F Spec R',
X* = {I~ • X I l~ is [-invariant},
Im/a = Im cr = {1, a}.
x'(o')
Since every prime ideal of a Boolean ring is maximal, it follows that Iu and Io are maximal ideals of R. So I, ~ Io, since kt 4: o. Choose x • 1~ a n d x $ I o . Then e - x • I o and e - x $ 1 v . Now, o(x) = / t ( e - x ) = tr and/~(x) = 1 = o(e - x ) . Let t • ] o ~ , l [ . Then ( x t ) ( x ) = t > o c = t r ( x ) , so that x, ¢ o. Hence o e X(x,). Also,
(e - x)t(e - x) = t > o: = t~(e - x), so that Further,
151
(e-x),¢/a.
Hence
I~•X((e-x),).
= x'
- v(o'),
where O' is any fuzzy ideal of R ', and g be a map from X ' to X* defined by g ( u ' ) - - f - ' ( t ~ ' ) , It' • X ' . Then: (i) g is continuous, (ii) g is open, and (iii) g is a homeomorphism of X ' onto X*.
Proof. (i): Let #' • X'. It follows from T h e o r e m 4.2 that f-~(/~') • X. Also f I(~LLe)is f-invariant, since for all a , b • R , if f ( a ) = f ( b ) then /~'(f(a)) = / ~ ' ( f ( b ) ) and thus
X(x,) N X ( ( e - x ) , ) = X ( ( x - xZ)t) by T h e o r e m 3.4(i) = X((0),)
since R is Boolean
=0. Consequently, A is Hausdorff.
(f-t(#'))(a) = (f '(g'))(b). Hence g(l~').=f-l(Iz') e X * . Next, let X(xl~ ) (3 X*, f l • ] O , 1], be any basic open set in X*. Then it is easy to verify
[] n x*) =
3.7. Theorem. If R is Boolean, o: • [0, 1[, and a = {/~ • X ]Im U = {1, a~}},
then the subspace A is compact, Hausdorff, and zero dimensional. Proof. Follows immediately 3.4(v) and 3.6(i), (iii). []
from
Theorems
This shows that the inverse image of any basic open set in X* is open in X'. Hence g is continuous. (ii) Let X'((f(x)t~), x e R and f l e ]0, 1], be any basic open set in X ' . Let o e g ( X ' ( ( f ( x ) ) t O ) . Then e = g ( / ~ ' ) = f - l ( ~ ') for some /~' • X ' such that (f(x))~ ~ l~'. As in part (i), we can show
R. Kumar / Fuzzy prime spectrum of a ring
152 that o is f-invariant. Next,
It follows from Theorem 4.5 that ~/it is a fuzzy semiprime ideal of R. The chain of level ideals of ~/it is given by
g(X'((f(x))~)) = X(x~) ¢3X*, because
\//~,~t[, i~_ \ / / i t t i C--- " " " 1~ \//ittm = R .
It • g(X'((f(x))t~)) ¢:> g-l(it) • X,((f(x))t~)
The following example illustrates the above definition.
and It is f-invariant
4.7. Example. Let p, m • Z+, where p is prime. Define a fuzzy subset It of Z (the ring of integers) by
¢:> f(xt~ ) = (f(x))a ¢fig-'(it) = f ( i t ) ¢:> x e q~f - ' ( f ( i t ) ) = It since It is f-invariant
It(x) = ¢,
It • X(x
) n x*.
Hence the direct image of every basic open set in X ' is open in X*; in consequence, g is open. (iii) In view of parts (i) and (ii), it is sufficient to show that g is one-one and onto. Let It', O' e X ' . Then g(it') = g ( O ' ) implies f-~(it') = f - l ( 0 ' ) which means f ( f - l ( # ' ) ) = f ( f - m ( o ' ) ) and thus I t ' = 0', since f is onto. Finally, let It z X*. Then It is an f-invariant fuzzy prime ideal of R and hence (by Theorem 4.2), f ( i t ) is a fuzzy prime ideal of R'. Further
{tti~ i f x • ( p m ) ' i f x • (pro-i>
(pro-i+1), i=1,2 .....
m,
where t i • ] 0 , 1 [ ( i = l . . . . . m) and t 0 > t m > • • • > tm. It is immediate from Theorem 2.2 that It is a fuzzy ideal of R. The chain of level ideals of It is given by
(p")
c
(pro-l)
C''"
C
(pO) = Z.
Since (pk) = ( p ) , K • Z+, it follows that
(\/it)(x)={tto
ifx • (p),
ifx•Z~(p).
g(f(it)) = f - a ( f ( i t ) ) = #, since It is f-invariant. Hence g is onto.
~
[] 4.8. Theorem [8]. Let 0 be any fuzzy ideal of
4.4. Definition [8]. A fuzzy ideal It of R is said to be fuzzy semiprime if It(X n) = It(X), X • R and
n•Z+. 4.5. Theorem [8]. A fuzzy ideal It of R is fuzzy semiprime iff each It,, t • Im It, is a semiprime ideal of R. (ii) A fuzzy prime ideal is always fuzzy semiprime. [] 4.6. Definition [6]. Let It be any fuzzy ideal of R with Im It = {to, tm. . . . .
tin}
and to > tl > • • • > tin. The fuzzy nil radical of It, symbolized by ~/it, is the fuzzy subset of R defined by to (k/it)(x)= ti
ifx • k/itt0,
ifx•X/it,,-X/it, .... i=1 . . . . . m,
R. Then: (i) X/0 is the smallest fuzzy semiprime ideal of R containing O. (ii) X/0 is the intersection of all fuzzy prime ideals of R containing O. [] 4.9. Theorem. Let 0 be any fuzzy ideal of R.
Then
(i)
V(O)=
V(~/O).
(ii) X(xt~)=X(yl~)
if and only if ~/(xt~) = ~/(y~ ), where x, y e R and [3 • ]0, 1].
Proof. (i): Let # • V(O). Since # is fuzzy prime (and hence fuzzy semiprime), it follows from Theorem 4.8(i) that ~/0 ___#. Hence # • V(V/O), so that V(O)~_ V(V~O). The reverse inclusion is obvious. (ii) If X(xt3)-- X(yt~ ) then V(x~)= V(ylO which implies V((x~))= V((yt~)). This means
where X/it,, denotes the nil radical of the ideal
~{#
It,, = {x
and therefore k/(x~) = ~/(Yt~)"
t¢ I It(x) >1 t,}.
] It e V((x~))} = ~ {it [ i t • V ( ( y t , ) ) }
R. Kumar / Fuzzy prime spectrum of a ring
any two n o n e m p t y basic open sets is nonempty. Hence, X i s irreducible. []
Conversely, let "X/(xt~) = k/(yt~ ). T h e n
#•V(xt~) ~
x~=_~ ¢~ (x~)=_~
¢:> k/(xt~) ~- U
153
by T h e o r e m 4.8(i)
5.3. T h e o r e m . The space X is disconnected iff R
has a nontrivial idempotent element. ¢:> yo~_#
as before
Proof. Let X be disconnected. Then there exist fuzzy ideals o and 0 of R such that
¢:> it • V(yt~). Hence
X(Yt,).
V(xt~ ) = V(ylO, []
so
that
X(xt~ ) =
X = V W ) U V(O), V(o)~=O,
5. Irreducibility and connectedness of the space X Recall that a topological space is irreducible iff the intersection of any two n o n e m p t y basic open sets is nonempty. Recall, too, that a topological space is disconnected iff it can be expressed as the union of two non-empty disjoint closed subsets.
V(O)~O,
V(o)nv(o)=~.
Now, V(o) n V(O) = ~ implies V ( o • 0) = ~ so that (o • O)(x) = 1 for all x • R. Hence sup (min(o(a), O(b))) = 1, e=a+b where
e is the
o(a)=O(b)=l e = a + b . Let
identity of R. This means for some a , b • R such that
I = {x • R ] o(x) = 1}, J={x•R
I O(x)= l},
5.1. Definition. The intersection of all the fuzzy prime ideals of R is called the fuzzy nil radical of R and is denoted by/~*.
P be any prime ideal of R, and # be the characteristic function of P. Then/~ • X. Since
5.2. T h e o r e m . The space X
it follows that o n 0 ~/~. Next, if x • 1 N J then (or fq O)(x) = 1 so that #(x) = 1 and thus x • P. This shows that every element of I n J is nilpotent. Now clearly,
is irreducible iff
It* • X . Proof. Let X be irreducible and let N be the nil radical of R. Then #*(x)=
1 0
ifx•N, ifx•R~N.
x = v ( o ) u v ( o ) = v ( o n o),
R / I N J =1/1 n J ~) J / I NJ. Therefore
e+lnJ=g+lNJ+h Next, let a, b ~_--R and let/3 • ]0, 1]. T h e n ab • N implies ab is nilpotent and thus X((ab)~) = ~, by T h e o r e m 3.4(ii). Therefore X(a/3) n X(b~) = f~, by T h e o r e m 3.4(i), which means X(al, ) = ~ or X(bl~ ) = ~, since X is irreducible. Hence either a or b is nilpotent, by T h e o r e m 3.4(ii), and thus a • N or b • N. Consequently, N is prime, whence it follows from T h e o r e m 2.4 that #* • X. Conversely, suppose that g* e X. Then N is pime. Let a , b • R and let f i e ] 0 , 1 ] . Then X(at~ ) NX(bl~ ) = ~ implies X((ab)~) =~, by T h e o r e m 3.4(i), and thus ab is nilpotent, by T h e o r e m 3.4(ii). Then ab • N and so a • N or b • N , which means a is nilpotent or b is nilpotent. Hence X(at~)=~ or X(bt~ ) = ~ , by T h e o r e m 3.4(ii). This shows that intersection of
+lNJ,
so that g 2 _ g • l n J and hence g 2 _ g is nilpotent. Thus (g2 g)m= 0 for some m • Z+. Consequently, gm =gm+lQ(g) for some polynomial Q(g) in g. Let x = gm(Q(g))m. It is a now simple matter to verify that x 4=0, x ~ e, and
X2~X. Conversely, for any nontrivial idempotent element x of R, it can be easily verified that
x = V(x~) u V((e - x)~), V(xt~)~,
v((e-x)t~)~,
V(xt~ ) n V((e - x)l~ ) = ~, where f l • ] 0 , 1]. This disconnected. []
establishes
that
X
is
154
R. Kumar / Fuzzy prime spectrum of a ring
Acknowledgements The author is thankful to the referees for their valuable comments and suggestions.
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