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Fuzzy invariant subgroups and fuzzy ideals
Fuzzy Sets and Systems 8 (1982) 133-139
133
North-Holland Publishing Company
FUZZY
INVARIANT
SUBGROLr~3
AND
FUZZY
IDEALS
Wang-jin L I U Deparmlent of Mathematics, Sichuan Teachers College, Sichuan, China Received March 1981 Revised May 1981 The purpose of this paper is to introduce some basic concepts of fuzzy algebra, as fuzzy invariant subgroups, fuzzy ideals, and to prove some fundamental properties. In particular, it will give a characteristic of a field by a fuzzy ideal.
Keywords: Fuzzy algebra.
Let X be the underlying set for a group. Fuzzy subgroups of X were defined by A. Rosenfeld in 1971 [1]. The purpose of this paper is to introduce some basic concepts of fuzzy algebra, as fuzzy invariant subgroups, quotient groups of X for fuzzy invariant subgroups, fuzzy subrings, fuzzy ideals, and to prove some fundamental properties. In particular, Proposition 3.4 will give a characteri.,fic of a (usual) field by a fuzzy ideal.
1. Preliminaries Let us recall some concepts occurring in the papers [1, 2, 3], which will be needed in the sequel. In this paper X always denotes a non-empty (usual) set. A fuzzy set in X is a map A:X--> [0, 1], and cg will denote the family of all fuzzy sets in X. Definition 1.1. Let '.' be a binary operation in X, and A, B ~ ~. Then the product A o B is defined as follows:
f
sup min(A(y), ( A o B ) ( x ) =~[0 "z=~
B(z) for y, z e X , y . z = x , for any y, z e X, y • z ¢ x.
It is clear that A o B ~ c6'
Proposition 1.1.
Let ,o' be as above, x~, y~, A, B ~ ~, where x~,, y~ are fuzzy
singletons, i.e.
x~(z)={0
ifz=x, if z e x,
y,,(z)={O
ifz=y, if z # y
0165-011418210000-00001502.75 © 1982 North-Holland
134
W.-Z Liu
for any z c X , and 0 < A, It <~ 1. Then: (i) x~ o y,, = ( x . y)m~,¢A,~,); (ii) A oB = I,Jx~A.v,,~a XAoy~,, where x~ c A ¢0 z~, ~ A .
Proof. (i) is clear from Definition 1.1. (ii) W e now take any point w ~ X and may assume there exist u, v e X, so that u • v = w, and A (u) > 0, B (v) > 0 without loss of generality. Firstly, ( A o B ) ( w ) = sup m i n ( A ( u ) , B(v))
>i
sup
min(xA(u), y~(v))
u-v = w.xxeA.¥,~eB
xxEA. ¥,~B
Secondly, since UAt,~ e A, vat~ e B, xx o y~,)(w)=
sup
sup min(x~(u), y,(v))
XxE.~k, V ~ E B U ' V = w
sup min(uA~u~(u), v~.~(v)) tl oD-----W
= sup min(A(u), B ( v ) ) M'I' =~lt
= ( A o B)(w).
Thus (A o B)(w) = ([.Jx,~A.y~Ea Xx o y~)(w). The following proposition holds from Definition 1.1. Proposition 1.2. Let 'o' be as above. (i) If the operation '.' in X is associative, commutative respectively, then so is 'o' in c~. (fi) I f the operation '.' in X has.a unit e, then the f u z z y singleton e ~ qg is a unit of the operation 'o' in c¢,, i.e. A o e = A = e o A for any A ~ c¢. Definition 1.2. Let X be a groupoid, i.e. a n o n - e m p t y set closed under a binary operation '-', A e qg, A ¢ ~. A is called a f u z z y subgroupoid, if AoAc_A.
The proofs of Propositions 1.3, 1.4 are omitted. Proposition 1.3. Let A ~ ~, A ~ ~. The following statements are equivalent: (i) A is a f u z z y subgroupoid. (ii) For any xx, y~, ~ A , then x~ o y~, ~ A . (iii) For any x, y ~ X, then A ( x . y)>~min(A(x), A(y)).
Fuzzy invariant subgroups and fuzzy ideals
135
ihroposilion 1.4. Let A ~ ~ be a f u z z y subgroupoid. (i) I[ the operation '.' in X is associative, then so is 'o' in A, i.e. for x~, y,, z, ~ A,
o(y. oz~).
( x~ o y~ ) o z~ = x~
(ii) I f the operation '.' in X is commutative, then so is 'o' in A, i.e. for x,,, y~ ~ A, xx o y~, = y~ o xA. (iii) I[ the operation :." in X has a unit e, then e o x~ = xa = x,, o e
]'or an y xx ~ A.
Now let X and Y be two groupoids, ]" is a (usual) homomorphism of X onto Y, i.e. ]': X --~ Y is a function onto, and for x, ~ ~ X, f ( x - ~) = ]'(x). f (z~). Let A ~ ~ ( X ) , B~q~(Y), then ]'(A), the image of A under [, is a fuzzy set in
~(Y): f sup A ( x )
if ] : - l ( y ) ~ 0 , if f-~(y) = O,
(f(A))(Y)=~O")=Y
for all y ~ Y, and f - l ( B ) , the preimage of B under f, is a fuzzy set in ~ ( X ) : ([-~(B))(x) = B ( f ( x ) ) ,
for all r c X
(of. [3, Definition 1.1.]). We have: (1) For any xx, Y& ~ ( X ) , f(x,.
o ~.) = (/(x)k o (f(~))..
Indeed, for all y ~ Y, if(x,, o
Jz.))(y)=
f((x
.
~)mi.,~..,)(Y)
= sup (x" f(z)=y
X)min(X,p.)(Z)
= { O in(A, 1~) if y = f ( x . 2), if y ¢ f ( x . Yc), ((f(x))x o (/(2))~,)(y) ( sup min((f(x))~(u), (f(~)).(v)
if u , v ~ Y , u ' v = y , if any u, v ~ Y, u - v -Y:y
if y ¢=f ( x ) .
f(Y¢).
(2) If A is a fuzzy subgroupoid oi X, then
f(A) is a fuzzy subgroupoid of Y.
Indeed, if f ( A ) is not a fuzzy subgroupoid, we have y, ~ ~ Y. so that
(f(A))(y. Y) < min((f(A))(y), (f(A))(9)),
W.-Z Liu
136
i.e. sup
A ( z ) < m i n ( sup A ( x ) , sup
f(z)=,j -9
Xf(x)=y
A(~)).
f(~)=~
Since f is onto, we now take x, ~ E X, such that [ ( x ) = y, f ( ~ ) = ~ and sup
A(z)
f(z)=y-~
Then A ( x . £)<- sup
A(z)
f(z)=y-~
which is impossible, since A is a fuzzy subgroupoid of X. (3) If B is a fuzzy subgroupoid, then so is f-~(B) (see [1]).
2. Fmmy invariant subgmeps D ~ n 2.1 [1]. If X is a group, A ~ c~, A ¢ O. A is called a [uzzy subgroup of x , if (i) A is a fuzzy subgroupoid of X, (ii) A ( x - l ) ~ A ( x ) for all x ~ X. Defudlion 2.2. Let A be a fuzzy subgroup of X; A will be called a fuzzy invariant subgroup of X, if A ( x • y) = A(y • x),
for any x, y ~ X.
For example, if X is a commutative group, then every fuzzy subgroup of X is a fuzzy invariant subgroup. lffolmsition 2.1. Let A be a [uzzy invariant subgroup of X. (i) For any B c qg, then A o B = B o A. (ii) If B is a fuzzy subgroup of X, then so is B o A.
lh~ot. (i) is clear from Definition 2.2. (ii) Firstly,
(BoA)o(B oA)=B
o ( A o B ) o A = B o(B o A ) o A
=(BoB)o(A oA)c_B oA. Secondly, for all x ~ X, (B o A)(x -~) =
sup
y.Z=X -!
=
min(B(y),
sup z-l-~]--I= g
>i
sup Z-t.y-l=
g
A(z))
min(B((y-1)-l), A((z-1)-I)) min(B(~'-l), A ( z - 1 ) ) = (A o B ) ( x ) = (B o A)(x).
Hence B o A is a fuzzy subgroup of X from Definition 2.1.
Fuzzy invariant subgroups and fuzzy ideals
137
Now let e e X be a unit of the operation '-' in X, and A e c~, A ¢: O, the symbol XA denotes the subset
{x ~ X iA(x)= A(e)}. Proposition 2.2. If A is a fuzzy invariant subgroup of X, then X,x is a invariant
subgroup of X. Fa'oof. It is clear that X,x ~ O. If x ~ XA, y ~ X, we have A ( y - x . y - l ) = A ( ( y - x)- y - ' ) = A ( y - ' - ( y . x))= A ( x ) = A(e), i.e. y • x • y - l e X A . Definition 2.3. Let A be a fuzzy invariant subgroup of X, then the quotient group X/XA is called the fuzzy quotient group of X with respect to the fuzzy invariant subgroup A.
Remark. If A ts a (usual) invariant subgroup of X, it is clear that A is a fuzzy invariant subgroup of X, and XA = A. NOW let j:X--> X/XA be the natural projection; we have:
Proposition 2.3. If A is a fuzzy invariant group of X, B e q¢, then j-J(j(B))= XA oB. Proof. For all x ~ X, (j-t(j(B)))(x) = (j(B))(j(x))=
sup
sup
B(y)=
j(y)=i(x)
x-y
B(y),
I~X^
and
(XAoB)(x)= sup min(XA(z),B(y))= z-y=x
sup
B(y).
z =x-y-tEXA
Let X and Y be two groups, and let f: x - - , ( be a group homomorphism (onto); we have: If A is a fuzzy subgroup of X, then so is f(A) in Y. Indeed, it suffices to show that (f(A))(y-')
=
A(z)=
sup f(z)=y
~
i> sup f(z-~)=y
sup
A((z-')-')
f(z-b=y
A ( z - l ) = sup A(u)=~f(A))(y) f(u)=v
for all y e Y. Similarly, if B is a f u r y subgroup of Y, then so is f-~'(B) in X (see Proposition 5.8 of [1]).
138
W.-J. Liu
3. Fuzzy ideals Definition 3.1. Let X be a ring with respect to two binary operations ' + ' , "-', and A e ~, A ~- ¢. A will be called a fuzzy subring of X, if A is a fuzzy subgroup for the binary operation ' + ' in qg induced b y ' + ' in X, and A is a fuzzy subgroupoid for the binaD operation 'o' in ~ induced by '.' in X. It is clear that subrings of X are fuzzy subrings of X. I[hroposilion 3.1. Let X be a ring, A E 9g, A ¢ ¢. Then A is a [uzzy subring of X if[, [or all x, y in X, (i) A ( x - y ) > ~ m i n ( A ( x ) , A ( y ) ) , (ii) A ( x - y ) > ~ m i n ( A ( x ) , A ( y ) ) .
Proof. This proposition follows directly from Proposition 5.6 in [1] and Proposition 1.3. Definition 3.2. Let X be a ring; a fuzzy subring A will be called a [uzzy le[t ideal of X, if A ( x . y)>~ A(y) for all x, y in X. It will be called a fuzzy right ideal, ff A ( x . y)~> A(x) for all x, u in X; and a [uzzy ideal, if it is a fuzzy left and right ide:L Now let X and Y be two rings, and let [: X---, Y be a ring homomorphism (onto). If A is a fuzzy subring (ideal) of X, then so is f(A); and if B is a f u r y subring (ideal) of Y, then so is f - t (B). The following proposition can be directly verified.
Proposition 3.2. Let X be a ring, A ~ ~, A ~: ¢. Then A is a [uzzy (left, right) ideal of X, iff for all x, y in X, (i) A ( x - y ) > ~ m i n ( A ( x ) , A ( y ) ) , (ii) A ( x . y)~>max(A(x), A ( y ) ) ( A ( x . y)>~ A ( y ) or A ( x . y)>~ A(x)). Proposition 3.3. Let X be a skew field (also division ring), A ~ ~¢, A ~: ¢. Then A is a fuzzy (le[t, right) ideal o [ X i f f A ( x ) = A(e)<~A(O) ]'or all x e X , x#O, where 0 is a unit of X for "+', e is a unit o[ X for '-'. Proof. Necessity. Since A (0) = A (e - e) >I min(A (e), A (e)) = A (e), and if x e X, x # O, then
A ( x ) = A ( x . e)>~ A(e), A ( e ) = A ( x -1 • x)>~ A(x). It follows that A ( x ) = A(e)<~ A(O). Su~ciency. (i) For all x, y in X, if x # y, then
A ( x - y) = A(e)>~min(A(x), A(y)),
Fuzzy invariant subgroups and fuzzy ideals
139
and if x = y, then
A ( x - y ) = A(O)>~min(A(x), A(y)). (ii) For all x, y in X, if x = 0 or y = 0, then
A ( x . y ) ~ m a x ( A ( x ) , A(y)) is clear, and if x ¢: 0, y ¢: 0, then
A(x . y)= A(e)=max(A(x),A(y)). This proves that A is a
fuzzy ideal.
Remark. The proposition shows that a fuzzy left (right) ideal is a fuzzy ideal in a skew field. The following proposition will give a characteristic of a (usual) field by a fuzzy ideal.
Proposition 3.4. Let X be a commutative ring with a unit e. Suppose for any fuzzy ideal A of X, A ( x ) = A ( e ) < ~ A ( O ) , x e X , x¢:O. Then X is a field.
Proof. Let A be a (usual) ideal, and A ¢: X. There exists ~ c X, ~¢ A; then A ( ~ ) = 0 , and A ( x ) = 0 for all x e X , x¢:0, therefore A - { 0 } . This proves that X is a field (see [4, Theorem 2.2, p. 100]).
References [1] A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl. 35 (1971j 512-517. [2] P.-M. Pu and Y.-M. Liu, Fuzz~ topology. 1. Neighborhood structure of a fuzzy point ~nd Moore-Smith convergence, J. Math. Anal. Appl. 76 (1980) 571-599, [3] P.-M. Pu and Y.-M. Liu, Fuzzy topology. It. Product and quotient spaces, J. M~th. Anal. Appl. 77 (1980) 20-3,7. [4] N. Jacobson, Basic Algebra 1 (Freeman, San Francisco, CA, 19741.
133
North-Holland Publishing Company
FUZZY
INVARIANT
SUBGROLr~3
AND
FUZZY
IDEALS
Wang-jin L I U Deparmlent of Mathematics, Sichuan Teachers College, Sichuan, China Received March 1981 Revised May 1981 The purpose of this paper is to introduce some basic concepts of fuzzy algebra, as fuzzy invariant subgroups, fuzzy ideals, and to prove some fundamental properties. In particular, it will give a characteristic of a field by a fuzzy ideal.
Keywords: Fuzzy algebra.
Let X be the underlying set for a group. Fuzzy subgroups of X were defined by A. Rosenfeld in 1971 [1]. The purpose of this paper is to introduce some basic concepts of fuzzy algebra, as fuzzy invariant subgroups, quotient groups of X for fuzzy invariant subgroups, fuzzy subrings, fuzzy ideals, and to prove some fundamental properties. In particular, Proposition 3.4 will give a characteri.,fic of a (usual) field by a fuzzy ideal.
1. Preliminaries Let us recall some concepts occurring in the papers [1, 2, 3], which will be needed in the sequel. In this paper X always denotes a non-empty (usual) set. A fuzzy set in X is a map A:X--> [0, 1], and cg will denote the family of all fuzzy sets in X. Definition 1.1. Let '.' be a binary operation in X, and A, B ~ ~. Then the product A o B is defined as follows:
f
sup min(A(y), ( A o B ) ( x ) =~[0 "z=~
B(z) for y, z e X , y . z = x , for any y, z e X, y • z ¢ x.
It is clear that A o B ~ c6'
Proposition 1.1.
Let ,o' be as above, x~, y~, A, B ~ ~, where x~,, y~ are fuzzy
singletons, i.e.
x~(z)={0
ifz=x, if z e x,
y,,(z)={O
ifz=y, if z # y
0165-011418210000-00001502.75 © 1982 North-Holland
134
W.-Z Liu
for any z c X , and 0 < A, It <~ 1. Then: (i) x~ o y,, = ( x . y)m~,¢A,~,); (ii) A oB = I,Jx~A.v,,~a XAoy~,, where x~ c A ¢0 z~, ~ A .
Proof. (i) is clear from Definition 1.1. (ii) W e now take any point w ~ X and may assume there exist u, v e X, so that u • v = w, and A (u) > 0, B (v) > 0 without loss of generality. Firstly, ( A o B ) ( w ) = sup m i n ( A ( u ) , B(v))
>i
sup
min(xA(u), y~(v))
u-v = w.xxeA.¥,~eB
xxEA. ¥,~B
Secondly, since UAt,~ e A, vat~ e B, xx o y~,)(w)=
sup
sup min(x~(u), y,(v))
XxE.~k, V ~ E B U ' V = w
sup min(uA~u~(u), v~.~(v)) tl oD-----W
= sup min(A(u), B ( v ) ) M'I' =~lt
= ( A o B)(w).
Thus (A o B)(w) = ([.Jx,~A.y~Ea Xx o y~)(w). The following proposition holds from Definition 1.1. Proposition 1.2. Let 'o' be as above. (i) If the operation '.' in X is associative, commutative respectively, then so is 'o' in c~. (fi) I f the operation '.' in X has.a unit e, then the f u z z y singleton e ~ qg is a unit of the operation 'o' in c¢,, i.e. A o e = A = e o A for any A ~ c¢. Definition 1.2. Let X be a groupoid, i.e. a n o n - e m p t y set closed under a binary operation '-', A e qg, A ¢ ~. A is called a f u z z y subgroupoid, if AoAc_A.
The proofs of Propositions 1.3, 1.4 are omitted. Proposition 1.3. Let A ~ ~, A ~ ~. The following statements are equivalent: (i) A is a f u z z y subgroupoid. (ii) For any xx, y~, ~ A , then x~ o y~, ~ A . (iii) For any x, y ~ X, then A ( x . y)>~min(A(x), A(y)).
Fuzzy invariant subgroups and fuzzy ideals
135
ihroposilion 1.4. Let A ~ ~ be a f u z z y subgroupoid. (i) I[ the operation '.' in X is associative, then so is 'o' in A, i.e. for x~, y,, z, ~ A,
o(y. oz~).
( x~ o y~ ) o z~ = x~
(ii) I f the operation '.' in X is commutative, then so is 'o' in A, i.e. for x,,, y~ ~ A, xx o y~, = y~ o xA. (iii) I[ the operation :." in X has a unit e, then e o x~ = xa = x,, o e
]'or an y xx ~ A.
Now let X and Y be two groupoids, ]" is a (usual) homomorphism of X onto Y, i.e. ]': X --~ Y is a function onto, and for x, ~ ~ X, f ( x - ~) = ]'(x). f (z~). Let A ~ ~ ( X ) , B~q~(Y), then ]'(A), the image of A under [, is a fuzzy set in
~(Y): f sup A ( x )
if ] : - l ( y ) ~ 0 , if f-~(y) = O,
(f(A))(Y)=~O")=Y
for all y ~ Y, and f - l ( B ) , the preimage of B under f, is a fuzzy set in ~ ( X ) : ([-~(B))(x) = B ( f ( x ) ) ,
for all r c X
(of. [3, Definition 1.1.]). We have: (1) For any xx, Y& ~ ( X ) , f(x,.
o ~.) = (/(x)k o (f(~))..
Indeed, for all y ~ Y, if(x,, o
Jz.))(y)=
f((x
.
~)mi.,~..,)(Y)
= sup (x" f(z)=y
X)min(X,p.)(Z)
= { O in(A, 1~) if y = f ( x . 2), if y ¢ f ( x . Yc), ((f(x))x o (/(2))~,)(y) ( sup min((f(x))~(u), (f(~)).(v)
if u , v ~ Y , u ' v = y , if any u, v ~ Y, u - v -Y:y
if y ¢=f ( x ) .
f(Y¢).
(2) If A is a fuzzy subgroupoid oi X, then
f(A) is a fuzzy subgroupoid of Y.
Indeed, if f ( A ) is not a fuzzy subgroupoid, we have y, ~ ~ Y. so that
(f(A))(y. Y) < min((f(A))(y), (f(A))(9)),
W.-Z Liu
136
i.e. sup
A ( z ) < m i n ( sup A ( x ) , sup
f(z)=,j -9
Xf(x)=y
A(~)).
f(~)=~
Since f is onto, we now take x, ~ E X, such that [ ( x ) = y, f ( ~ ) = ~ and sup
A(z)
f(z)=y-~
Then A ( x . £)<- sup
A(z)
f(z)=y-~
which is impossible, since A is a fuzzy subgroupoid of X. (3) If B is a fuzzy subgroupoid, then so is f-~(B) (see [1]).
2. Fmmy invariant subgmeps D ~ n 2.1 [1]. If X is a group, A ~ c~, A ¢ O. A is called a [uzzy subgroup of x , if (i) A is a fuzzy subgroupoid of X, (ii) A ( x - l ) ~ A ( x ) for all x ~ X. Defudlion 2.2. Let A be a fuzzy subgroup of X; A will be called a fuzzy invariant subgroup of X, if A ( x • y) = A(y • x),
for any x, y ~ X.
For example, if X is a commutative group, then every fuzzy subgroup of X is a fuzzy invariant subgroup. lffolmsition 2.1. Let A be a [uzzy invariant subgroup of X. (i) For any B c qg, then A o B = B o A. (ii) If B is a fuzzy subgroup of X, then so is B o A.
lh~ot. (i) is clear from Definition 2.2. (ii) Firstly,
(BoA)o(B oA)=B
o ( A o B ) o A = B o(B o A ) o A
=(BoB)o(A oA)c_B oA. Secondly, for all x ~ X, (B o A)(x -~) =
sup
y.Z=X -!
=
min(B(y),
sup z-l-~]--I= g
>i
sup Z-t.y-l=
g
A(z))
min(B((y-1)-l), A((z-1)-I)) min(B(~'-l), A ( z - 1 ) ) = (A o B ) ( x ) = (B o A)(x).
Hence B o A is a fuzzy subgroup of X from Definition 2.1.
Fuzzy invariant subgroups and fuzzy ideals
137
Now let e e X be a unit of the operation '-' in X, and A e c~, A ¢: O, the symbol XA denotes the subset
{x ~ X iA(x)= A(e)}. Proposition 2.2. If A is a fuzzy invariant subgroup of X, then X,x is a invariant
subgroup of X. Fa'oof. It is clear that X,x ~ O. If x ~ XA, y ~ X, we have A ( y - x . y - l ) = A ( ( y - x)- y - ' ) = A ( y - ' - ( y . x))= A ( x ) = A(e), i.e. y • x • y - l e X A . Definition 2.3. Let A be a fuzzy invariant subgroup of X, then the quotient group X/XA is called the fuzzy quotient group of X with respect to the fuzzy invariant subgroup A.
Remark. If A ts a (usual) invariant subgroup of X, it is clear that A is a fuzzy invariant subgroup of X, and XA = A. NOW let j:X--> X/XA be the natural projection; we have:
Proposition 2.3. If A is a fuzzy invariant group of X, B e q¢, then j-J(j(B))= XA oB. Proof. For all x ~ X, (j-t(j(B)))(x) = (j(B))(j(x))=
sup
sup
B(y)=
j(y)=i(x)
x-y
B(y),
I~X^
and
(XAoB)(x)= sup min(XA(z),B(y))= z-y=x
sup
B(y).
z =x-y-tEXA
Let X and Y be two groups, and let f: x - - , ( be a group homomorphism (onto); we have: If A is a fuzzy subgroup of X, then so is f(A) in Y. Indeed, it suffices to show that (f(A))(y-')
=
A(z)=
sup f(z)=y
~
i> sup f(z-~)=y
sup
A((z-')-')
f(z-b=y
A ( z - l ) = sup A(u)=~f(A))(y) f(u)=v
for all y e Y. Similarly, if B is a f u r y subgroup of Y, then so is f-~'(B) in X (see Proposition 5.8 of [1]).
138
W.-J. Liu
3. Fuzzy ideals Definition 3.1. Let X be a ring with respect to two binary operations ' + ' , "-', and A e ~, A ~- ¢. A will be called a fuzzy subring of X, if A is a fuzzy subgroup for the binary operation ' + ' in qg induced b y ' + ' in X, and A is a fuzzy subgroupoid for the binaD operation 'o' in ~ induced by '.' in X. It is clear that subrings of X are fuzzy subrings of X. I[hroposilion 3.1. Let X be a ring, A E 9g, A ¢ ¢. Then A is a [uzzy subring of X if[, [or all x, y in X, (i) A ( x - y ) > ~ m i n ( A ( x ) , A ( y ) ) , (ii) A ( x - y ) > ~ m i n ( A ( x ) , A ( y ) ) .
Proof. This proposition follows directly from Proposition 5.6 in [1] and Proposition 1.3. Definition 3.2. Let X be a ring; a fuzzy subring A will be called a [uzzy le[t ideal of X, if A ( x . y)>~ A(y) for all x, y in X. It will be called a fuzzy right ideal, ff A ( x . y)~> A(x) for all x, u in X; and a [uzzy ideal, if it is a fuzzy left and right ide:L Now let X and Y be two rings, and let [: X---, Y be a ring homomorphism (onto). If A is a fuzzy subring (ideal) of X, then so is f(A); and if B is a f u r y subring (ideal) of Y, then so is f - t (B). The following proposition can be directly verified.
Proposition 3.2. Let X be a ring, A ~ ~, A ~: ¢. Then A is a [uzzy (left, right) ideal of X, iff for all x, y in X, (i) A ( x - y ) > ~ m i n ( A ( x ) , A ( y ) ) , (ii) A ( x . y)~>max(A(x), A ( y ) ) ( A ( x . y)>~ A ( y ) or A ( x . y)>~ A(x)). Proposition 3.3. Let X be a skew field (also division ring), A ~ ~¢, A ~: ¢. Then A is a fuzzy (le[t, right) ideal o [ X i f f A ( x ) = A(e)<~A(O) ]'or all x e X , x#O, where 0 is a unit of X for "+', e is a unit o[ X for '-'. Proof. Necessity. Since A (0) = A (e - e) >I min(A (e), A (e)) = A (e), and if x e X, x # O, then
A ( x ) = A ( x . e)>~ A(e), A ( e ) = A ( x -1 • x)>~ A(x). It follows that A ( x ) = A(e)<~ A(O). Su~ciency. (i) For all x, y in X, if x # y, then
A ( x - y) = A(e)>~min(A(x), A(y)),
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and if x = y, then
A ( x - y ) = A(O)>~min(A(x), A(y)). (ii) For all x, y in X, if x = 0 or y = 0, then
A ( x . y ) ~ m a x ( A ( x ) , A(y)) is clear, and if x ¢: 0, y ¢: 0, then
A(x . y)= A(e)=max(A(x),A(y)). This proves that A is a
fuzzy ideal.
Remark. The proposition shows that a fuzzy left (right) ideal is a fuzzy ideal in a skew field. The following proposition will give a characteristic of a (usual) field by a fuzzy ideal.
Proposition 3.4. Let X be a commutative ring with a unit e. Suppose for any fuzzy ideal A of X, A ( x ) = A ( e ) < ~ A ( O ) , x e X , x¢:O. Then X is a field.
Proof. Let A be a (usual) ideal, and A ¢: X. There exists ~ c X, ~¢ A; then A ( ~ ) = 0 , and A ( x ) = 0 for all x e X , x¢:0, therefore A - { 0 } . This proves that X is a field (see [4, Theorem 2.2, p. 100]).
References [1] A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl. 35 (1971j 512-517. [2] P.-M. Pu and Y.-M. Liu, Fuzz~ topology. 1. Neighborhood structure of a fuzzy point ~nd Moore-Smith convergence, J. Math. Anal. Appl. 76 (1980) 571-599, [3] P.-M. Pu and Y.-M. Liu, Fuzzy topology. It. Product and quotient spaces, J. M~th. Anal. Appl. 77 (1980) 20-3,7. [4] N. Jacobson, Basic Algebra 1 (Freeman, San Francisco, CA, 19741.