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Invertibility of morphisms in the category of L-fuzzy left R-modules
ZZY
sets and systems ELSEVIER
Fuzzy Sets and Systems 105 (1999) 503-507
Short Communication
Invertibility ofmorphisms in the category of L-fuzzy left R-modules Jiangang Tang Department of Mathematics, YiLi Teachers Colleye, ginin~l City, Xinjiang, People's Republic of China Received August 1997; received in revised form May 1998
Abstract In this paper, we obtain the sufficient and necessary condition for a left (right) invertible morphism in the category of L-fuzzy left R-modules, and point out a mistake made in Fuzzy Sets and Systems 21 (1987) 105-113. (~) 1999 Elsevier Science B.V. All rights reserved.
Keywords: L-fuzzy left R-module; Category; Left invertible morphism; Right invertible morphism
1. Introduction Since fuzzy modules were introduced by Negoita and Ralescu [6], Katsaras and Liu [2] and Lowen [3] have developed the theory of fuzzy vector spaces, Pan [7] introduced the category of fuzzy modules and discussed the construction of fuzzy finitely generated modules. Tang [9-11] introduced the category of L-fuzzy left R-modules and discussed product, coproduct, free object, tensor product in the category of L-fuzzy left R-modules. In this paper, we study the invertibility of morphism in the category of L-fuzzy left R-modules, and obtain the sufficient and necessary condition for a left (right) invertible morphism, we give a counterexample to show that the condition given in the Theorem of [7] is not sufficient.
non-empty (usual) set. An L-fuzzy set of X is a map A :X--oL, and L x will denote the family of all L-fuzzy sets in X. Then Lx is a completely distributive lattice in a natural manner, and the least and greatest elements of L x are denoted by 0 and 1, respectively, where 0(x) = 0, l ( x ) = 1 for any x EX. Let R be a ring, M, N, P be the three left R-modules which have zero elements 0M, ON, 0e, respectively.
2. Preliminary notions
Definition 2.1. Let A ELM; A will be called an Lfuzzy left R-submodule of M, if the following hold: (1) A(0M) = l; (2) for any x, y E M: A(x)/~ A(y) <.A(x + y); (3) for any x EM: A(x)<.A(-x); (4) for any x E M and r E R: A(x) <.A(rx). The pair (M,A) is called an L-fuzzy left R-module or an L-tim for short. We always let F ( M ) denote the family of all L-fuzzy left R-submodules of M.
Throughout this paper L will denote a completely distributive lattice, which has the least and greatest element, say 0 and 1 respectively. Suppose that X is a
Remark 1. From (2) and (3) of Definition 2.1 we obtain A(0M) = V { A ( x ) l x E M}, so conditions ( 1 ) - ( 3 ) are consistent.
0165-0114/99/$ - see front matter (~) 1999 Elsevier Science B.V. All rights reserved. PII: S0165-0114(98)00282-6
J. Tana/Fuzzy Sets and Systems 105 (1999) 503-507
504
Remark 2. IfL : [0, 1], the L-fuzzy left R-module is the same as that of [7], so Definition 2.1 generalizes the concept of fuzzy module in [7]. Definition 2.2. Let (M,A) and (N, B) be two L-fuzzy left R-modules, an R-homomorphism (R-isomorphism) f : M ~ N is called an F-homomorphism (F-isomorphism) denoted by f : (M,A) ~ (N,B), if f ( A ) <~B(f(A) = B). The following propositions can be directly verified.
Proposition 2.1. Let (M,A),(N,B) be the L-juzzy left R-modules, f :M --* N & an R-homomorphism (R-isomorphism), then the following statements are equivalent, (1) f(A)<~B ( f ( A ) = B ) , (2) A <~Bf(A = Bf).
Proposition 2.2. Let f ' ( M , A ) ~ (N,B) be an Fhomomorphism and for any x E Ker f , ( f Ker f ) ( x ) = A(x), then f Ker f is an L-fuzzy left R-submodule of Ker f , where Ker f is a kernel of the R-homomorphism f ( K e r r , f K e r f ) is called the F-kernel of the F-homomorphism f Proposition 2.3. Let f : (M,A) ~ (N,B) be an Fhomomorphism and jbr any y E I m f , ( f Im f ) ( y ) = f ( A ) ( y ) , then f l m f is an L-fuzzy left Rsubmodule of Im f , where Im f is an image of the R-homomorphism f ( I m f , f l m f ) is called the F-image of the F-homomorphism f
Definition2.4. Let (M,A),(N,B) be the L-fuzzy left R-modules, (N,B) is called the F-submodule of (M,A), if N < M and B(x)<~A(x) for e v e n x E N . (M,A) and (N,B) are equal, denoted by (M,A) = (N,B), i f M = N andA = B .
Definition 2.5. Let (M,A) be an L-tim, and (N,B), (P, C) be the F-submodules of (M, A ), (M, A ) is called an F-sum of (N,B) and (P, C), denoted by (M,A) = (N,B) + (P, C), if (1)M=N+P; (2) A = B + C, i.e., for any x E M: A(x) = V{B(u) A c(v)luEN, v E P a n d u + v = x } . (M, A) is called a direct F-sum of(N, B) and (P, C), denoted by (M,A)=(N,B)O(P, C), i f ( l ) M = N O P; (2)A=B+C. Definition 2.6. We shall say that F-submodules (N,B) and (P, C) of an L-tim (M,A) are supplementary, if (M,A) = (N,B) • (P,C). An F-submodule (N,B) of (M,A) is called a direct summand of (M,A), if there is an F-submodule (P, C) of (M,A) such that (N,B) and (P, C) are supplementary.
3. The invertibility of morphism Definition 3.1. Suppose f E Hom((M,A), (N,B)), f is called a right (left) invertible, if there exists a g E nom((N, B), (M, A ) ) such that f 0 = 1N ( # f : 1M), f is called invertible, if it is both right and left invertible.
Defnition 2.3. The category of L-fuzzy left R-modules R-mod(L), is as follows: The family Ob(R-mod(L)) of objects is the family of all L-tim's, for any (M,A),(N,B)EOb(R-mod(L)), the set of morphisms is
Proposition 3.1. A morphism f E Hom ((M, A), (N,B)) is right invertible if and only if ( K e r f , f Ker f ) is a direct summand of(M, A) and f ( A ) = B and f is an R-epimorphism.
HOmR-mod(L)((M,A ), (N, B))
Proof. Necessity: If morphism f E Hom((M,A),
= { f l f : (M,A) ~ (N,B) is an F-homomorphism}, the composition of morphisms is the usual composition operation of mappings. Similar to [9 11], we can verify that R-mod(L) is indeed a category.
(N,B)) is right invertible, then there exists a gEHom((N,B),(M,A)) such that f9 = IN, SO f is an epimorphism. Consider K e r r A Img, if x E K e r f A Img, then f ( x ) = ON and x = g(Y) for some y E N, hence ON = f ( x ) = f ( g ( Y ) ) = Y and consequently x = g(Y) = g(0N) = 0 i . Thus we see that Ker f n Im g = {0M }. Moreover, for every x E M, we have f ( x - g(f(x))) = f ( x ) - (fgf)(x) = f ( x ) f ( x ) = ON and so x - (gf)(x) E Ker f Now, since
J. Tang~Fuzzy Sets and Systems 105 (1999) 503-507 every x E M can be written as x = ( g f ) ( x ) + x - ( g f ) ( x ) , we see that M = I m g + K e r f. Hence M = I m g@Ker f For any y E N, B(y)<~ A(g(y))<~ B ( f ( g ( y ) ) ) = B(y ) implies B = Ag, and for any x E M, B ( f (x ) ) <~
A ( g ( f (x) ) ) <~B ( f (,q(f (x) ) ) = B ( f (x ) ) ) implies B f = A.qf. For any x C M:
505
an R-isomorphism. Then for any y E N, since
f ( A ) ( y ) = V { A ( x ) [ x 6 M, f ( x ) = y} = V { A ( u ) A C ( v ) l u ~ K e r f . v C P, u + v = x, f ( x )
= v}
( f Im g + .1" Ker f ) ( x )
= V{(f
Im g)(u) A ( f Ker f ) ( v ) l
u~Img,
vEKerf,
u+v=x}.
= C(f~-I(y)) AA(0M)
<~A(.fp-~(y)),
Since ( f lm g)(u) = V { B ( s ) l g ( s ) = u, s E N}
= V { A ( g ( s ) ) l g ( s ) = u, s E N }
and f ( A ) = B, so B ~
and
R e m a r k 3. In Theorem 1.1 o f [7] the condition is not sufficient, we give a counterexample: Taking a prime number p, let
( f K e r f ) ( v ) = A(v),
Qp={xEQtx=k/p",
next, for any x E M, x can be written in a unique way as x = ( # f ) ( x ) + x - (gf)(x), then
= A(x) A B ( f ( x ) )
where Q is the set o f rational numbers, Z is the set of integers, N is the set of positive integers, then Qp is a subgroup of Q, Z is a subgroup of Q and Z is a subgroup of Qp, the quotient group Qp/Z is a Z-left module, let A :Qp/Z ---+L, A(x) --- 1, for any x 6 Qp/Z, then A is an L-fuzzy left R-submodule o f Qp/Z, where L = [0, 1], put f : Qp/Z --, Qp/Z where .f(x) = px, x 6 Qp/Z, then f is a Z-homomorphism obviously
= A(x),
f :(Qp/Z,A) ---+(Qp/Z,A)
=A(u),
(.f Im g + f Ker f ) ( x ) = A((fff)(x)) A A ( x - ( g f ) ( x ) )
>~ A((gf)(x)) A A(x) A A((gf)(x))
and ( f l m , q + f K e r f ) ( x ) < ~ A ( x ) , so f l m g + f K e r r = A Therefore (M,A) = (Img, f l m g ) • ( K e r f , f K e r f ) , implying ( K e r f , f K e r f ) is a direct summand of (M,A). Since B = f ( g ( B ) ) ~<.f(A) ~
kEZ, nCN},
is an F-homomorphism and for any y ~ Qp/Z:
A(y)=max{A(x)]xE f
i(y)},
so the condition of the Theorem 1.1 o f [7], holds but f is not right invertible ( f is not the fuzzy split in sense of [7]), otherwise, there exists an F-homomorphism
g :(Qp/Z,A) ---+(Qp/Z,A), such that J~/ = l Q,,/z, [ p - i ] denotes the equivalence class of p - i , then f ( g ( [ p - I ] ) ) = [ p - i ] , but . f ( g ( [ p - ~ ] ) ) = p ( g ( [ p - I ] ) ) = g(p[p-~]) = g([0]) = [0], thus [ p - f ] = [0], it is not possible, since p-~ EZ.
J. TanylFuzzy Sets and Systems 105 (1999) 503-507
506
Proposition 3.2. A
morphism f E Hom((M, A), (N,B)) is left invertible if and only if f l m f is a direct summand of (N,B) and A = Bf and f is an R-monomorphism. Proof. If morphism f E Hom((M,A),(N,B)) is left invertible, then there exists a 0 E Hom((N, B), (M,A)) such that 9f = 1M, therefore f is an R-monomorphism. Because A <~Bf<~Agf<<.A, so A = Bf. If y E I m f fq Ker 9, then y = f ( x ) for some x E M and 9(Y) = 0M, thus 0M = 9(Y) = 9(f(x)) = x, SO y = f ( x ) = f ( 0 M ) = 0N, hence I m f fq K e r 9 = {0}. Since every y E N can be written as y = (fg)(Y) + Y - (fg)(Y), where y - (fg)(Y) E Ker9, we deduce that N = I m f ® Ker o. Because A~t<~Bfo<<.Aofo = Ag, so A9 = Bf9. For every Y E N, since
by setting 9(Y)= 9(u + v)= f - l ( u ) , then for all x E M we have
(9f)(x) = g(f(x)) = f - I ( f ( x ) ) = x so that gf = 1M. For every y E N, since
A(O(y)) = A ( f - I ( u ) ) = B(f(f-l(u))) = B(u)
and
B(y) = ( f Im f ) ( u ) A C(v)
= (V{-.,(s)
( f l m f)((fo)(Y)) = V{A(s)ls E M, f(s) =
f(g)(y)}
= B(u)/~ C(v) <~ B(u),
H
I
_
_
,
= V t B ( / ( s ) ) l s EM, f(s)
=
(fg)(y)}
= B((fg)(y)) = A(g(y)) B(y). so
so B <<.Ay, therefore g E Hom((N,B), (M,A)). Since 9 f = Ira, we have f is a left invertible. []
Proposition 3.3. A
morphism f E Hom ((M, A), (N, B)) is invertible if and only if f is an R-isomorphism and A = B f (or f ( A ) = B). It is obvious.
(flmf
+ fKero)(y)
= ( f l m f ) ( ( f o ) ( y ) ) A ( ( f K e r o ) ( y - (fo)(Y)))
Acknowledgements
= B(y) A B(y - (fo)(Y))
I would like to express my thanks for the constructive and helpful comments o f two anonymous referees.
>>.B(y) A B(y) A B((fo)(y))
B(y) and f Im f + f Ker 9 ~
(N,B). If ( I m f , f I m f )
is a direct summand o f (N,B), then there exists an L-fuzzy left R-module (P, C) such that ( N , B ) = ( I m f , f I m f ) @ ( P , C) s o N = I m f ~ P, then for every y E N, y can be written uniquely as y = u + v, such that f ( x ) = u so we can write this element as f - I ( u ) without confusion. N o w define g:N --. M
References [1] N. Jacobson, Basic Algebra (11), Freeman, San Francisco, 1980. [2] A.K. Katsaras, D.B. Liu, Fuzzy vector spaces and fuzzy topological vector spaces, J. Math. Anal. Appl. 58 (1977) 135-146. [3] R. Lowen, Convex sets, Fuzzy Sets and Systems 3 (1980) 291-310. [4] W. Liu, Fuzzy invariant subgroups and fuzzy ideals, Fuzzy Sets and Systems 8 (1982) 133-139.
Z Tang/Fuzzy Sets and Systems 105 (1999) 503-507 [5] W. Liu, Operations on fuzzy ideals, Fuzzy Sets and Systems 11 (1983) 31-41. [6] C.V. Negoita, D.A. Ralescu, Application of Fuzzy Sets to System Analysis, Birkhauser, Basel, 1975. [7] F. Pan, Fuzzy finitely generated modules, Fuzzy Sets and Systems 21 (1987) 105-113. [8] A. Rosenfeld, Fuzzy group, J. Math. Anal. Appl. 35 (1971) 512-517.
507
[9] J. Tang, Free object for the category of L-fuzzy left R-modules determined by L-fuzzy sets, 5th IFSA World Congress, 1993, pp. 357-359. [10] J. Tang, Free L-fuzzy left R-module, Fuzzy Systems and Math. (China) 1 (1991) 23-26. [I 1] J. Tang, Tensor product and tensor functor of the category of L-fuzzy modules, Fuzzy Systems and Math. (China) 3 (1995) 65-73.
sets and systems ELSEVIER
Fuzzy Sets and Systems 105 (1999) 503-507
Short Communication
Invertibility ofmorphisms in the category of L-fuzzy left R-modules Jiangang Tang Department of Mathematics, YiLi Teachers Colleye, ginin~l City, Xinjiang, People's Republic of China Received August 1997; received in revised form May 1998
Abstract In this paper, we obtain the sufficient and necessary condition for a left (right) invertible morphism in the category of L-fuzzy left R-modules, and point out a mistake made in Fuzzy Sets and Systems 21 (1987) 105-113. (~) 1999 Elsevier Science B.V. All rights reserved.
Keywords: L-fuzzy left R-module; Category; Left invertible morphism; Right invertible morphism
1. Introduction Since fuzzy modules were introduced by Negoita and Ralescu [6], Katsaras and Liu [2] and Lowen [3] have developed the theory of fuzzy vector spaces, Pan [7] introduced the category of fuzzy modules and discussed the construction of fuzzy finitely generated modules. Tang [9-11] introduced the category of L-fuzzy left R-modules and discussed product, coproduct, free object, tensor product in the category of L-fuzzy left R-modules. In this paper, we study the invertibility of morphism in the category of L-fuzzy left R-modules, and obtain the sufficient and necessary condition for a left (right) invertible morphism, we give a counterexample to show that the condition given in the Theorem of [7] is not sufficient.
non-empty (usual) set. An L-fuzzy set of X is a map A :X--oL, and L x will denote the family of all L-fuzzy sets in X. Then Lx is a completely distributive lattice in a natural manner, and the least and greatest elements of L x are denoted by 0 and 1, respectively, where 0(x) = 0, l ( x ) = 1 for any x EX. Let R be a ring, M, N, P be the three left R-modules which have zero elements 0M, ON, 0e, respectively.
2. Preliminary notions
Definition 2.1. Let A ELM; A will be called an Lfuzzy left R-submodule of M, if the following hold: (1) A(0M) = l; (2) for any x, y E M: A(x)/~ A(y) <.A(x + y); (3) for any x EM: A(x)<.A(-x); (4) for any x E M and r E R: A(x) <.A(rx). The pair (M,A) is called an L-fuzzy left R-module or an L-tim for short. We always let F ( M ) denote the family of all L-fuzzy left R-submodules of M.
Throughout this paper L will denote a completely distributive lattice, which has the least and greatest element, say 0 and 1 respectively. Suppose that X is a
Remark 1. From (2) and (3) of Definition 2.1 we obtain A(0M) = V { A ( x ) l x E M}, so conditions ( 1 ) - ( 3 ) are consistent.
0165-0114/99/$ - see front matter (~) 1999 Elsevier Science B.V. All rights reserved. PII: S0165-0114(98)00282-6
J. Tana/Fuzzy Sets and Systems 105 (1999) 503-507
504
Remark 2. IfL : [0, 1], the L-fuzzy left R-module is the same as that of [7], so Definition 2.1 generalizes the concept of fuzzy module in [7]. Definition 2.2. Let (M,A) and (N, B) be two L-fuzzy left R-modules, an R-homomorphism (R-isomorphism) f : M ~ N is called an F-homomorphism (F-isomorphism) denoted by f : (M,A) ~ (N,B), if f ( A ) <~B(f(A) = B). The following propositions can be directly verified.
Proposition 2.1. Let (M,A),(N,B) be the L-juzzy left R-modules, f :M --* N & an R-homomorphism (R-isomorphism), then the following statements are equivalent, (1) f(A)<~B ( f ( A ) = B ) , (2) A <~Bf(A = Bf).
Proposition 2.2. Let f ' ( M , A ) ~ (N,B) be an Fhomomorphism and for any x E Ker f , ( f Ker f ) ( x ) = A(x), then f Ker f is an L-fuzzy left R-submodule of Ker f , where Ker f is a kernel of the R-homomorphism f ( K e r r , f K e r f ) is called the F-kernel of the F-homomorphism f Proposition 2.3. Let f : (M,A) ~ (N,B) be an Fhomomorphism and jbr any y E I m f , ( f Im f ) ( y ) = f ( A ) ( y ) , then f l m f is an L-fuzzy left Rsubmodule of Im f , where Im f is an image of the R-homomorphism f ( I m f , f l m f ) is called the F-image of the F-homomorphism f
Definition2.4. Let (M,A),(N,B) be the L-fuzzy left R-modules, (N,B) is called the F-submodule of (M,A), if N < M and B(x)<~A(x) for e v e n x E N . (M,A) and (N,B) are equal, denoted by (M,A) = (N,B), i f M = N andA = B .
Definition 2.5. Let (M,A) be an L-tim, and (N,B), (P, C) be the F-submodules of (M, A ), (M, A ) is called an F-sum of (N,B) and (P, C), denoted by (M,A) = (N,B) + (P, C), if (1)M=N+P; (2) A = B + C, i.e., for any x E M: A(x) = V{B(u) A c(v)luEN, v E P a n d u + v = x } . (M, A) is called a direct F-sum of(N, B) and (P, C), denoted by (M,A)=(N,B)O(P, C), i f ( l ) M = N O P; (2)A=B+C. Definition 2.6. We shall say that F-submodules (N,B) and (P, C) of an L-tim (M,A) are supplementary, if (M,A) = (N,B) • (P,C). An F-submodule (N,B) of (M,A) is called a direct summand of (M,A), if there is an F-submodule (P, C) of (M,A) such that (N,B) and (P, C) are supplementary.
3. The invertibility of morphism Definition 3.1. Suppose f E Hom((M,A), (N,B)), f is called a right (left) invertible, if there exists a g E nom((N, B), (M, A ) ) such that f 0 = 1N ( # f : 1M), f is called invertible, if it is both right and left invertible.
Defnition 2.3. The category of L-fuzzy left R-modules R-mod(L), is as follows: The family Ob(R-mod(L)) of objects is the family of all L-tim's, for any (M,A),(N,B)EOb(R-mod(L)), the set of morphisms is
Proposition 3.1. A morphism f E Hom ((M, A), (N,B)) is right invertible if and only if ( K e r f , f Ker f ) is a direct summand of(M, A) and f ( A ) = B and f is an R-epimorphism.
HOmR-mod(L)((M,A ), (N, B))
Proof. Necessity: If morphism f E Hom((M,A),
= { f l f : (M,A) ~ (N,B) is an F-homomorphism}, the composition of morphisms is the usual composition operation of mappings. Similar to [9 11], we can verify that R-mod(L) is indeed a category.
(N,B)) is right invertible, then there exists a gEHom((N,B),(M,A)) such that f9 = IN, SO f is an epimorphism. Consider K e r r A Img, if x E K e r f A Img, then f ( x ) = ON and x = g(Y) for some y E N, hence ON = f ( x ) = f ( g ( Y ) ) = Y and consequently x = g(Y) = g(0N) = 0 i . Thus we see that Ker f n Im g = {0M }. Moreover, for every x E M, we have f ( x - g(f(x))) = f ( x ) - (fgf)(x) = f ( x ) f ( x ) = ON and so x - (gf)(x) E Ker f Now, since
J. Tang~Fuzzy Sets and Systems 105 (1999) 503-507 every x E M can be written as x = ( g f ) ( x ) + x - ( g f ) ( x ) , we see that M = I m g + K e r f. Hence M = I m g@Ker f For any y E N, B(y)<~ A(g(y))<~ B ( f ( g ( y ) ) ) = B(y ) implies B = Ag, and for any x E M, B ( f (x ) ) <~
A ( g ( f (x) ) ) <~B ( f (,q(f (x) ) ) = B ( f (x ) ) ) implies B f = A.qf. For any x C M:
505
an R-isomorphism. Then for any y E N, since
f ( A ) ( y ) = V { A ( x ) [ x 6 M, f ( x ) = y} = V { A ( u ) A C ( v ) l u ~ K e r f . v C P, u + v = x, f ( x )
= v}
( f Im g + .1" Ker f ) ( x )
= V{(f
Im g)(u) A ( f Ker f ) ( v ) l
u~Img,
vEKerf,
u+v=x}.
= C(f~-I(y)) AA(0M)
<~A(.fp-~(y)),
Since ( f lm g)(u) = V { B ( s ) l g ( s ) = u, s E N}
= V { A ( g ( s ) ) l g ( s ) = u, s E N }
and f ( A ) = B, so B ~
and
R e m a r k 3. In Theorem 1.1 o f [7] the condition is not sufficient, we give a counterexample: Taking a prime number p, let
( f K e r f ) ( v ) = A(v),
Qp={xEQtx=k/p",
next, for any x E M, x can be written in a unique way as x = ( # f ) ( x ) + x - (gf)(x), then
= A(x) A B ( f ( x ) )
where Q is the set o f rational numbers, Z is the set of integers, N is the set of positive integers, then Qp is a subgroup of Q, Z is a subgroup of Q and Z is a subgroup of Qp, the quotient group Qp/Z is a Z-left module, let A :Qp/Z ---+L, A(x) --- 1, for any x 6 Qp/Z, then A is an L-fuzzy left R-submodule o f Qp/Z, where L = [0, 1], put f : Qp/Z --, Qp/Z where .f(x) = px, x 6 Qp/Z, then f is a Z-homomorphism obviously
= A(x),
f :(Qp/Z,A) ---+(Qp/Z,A)
=A(u),
(.f Im g + f Ker f ) ( x ) = A((fff)(x)) A A ( x - ( g f ) ( x ) )
>~ A((gf)(x)) A A(x) A A((gf)(x))
and ( f l m , q + f K e r f ) ( x ) < ~ A ( x ) , so f l m g + f K e r r = A Therefore (M,A) = (Img, f l m g ) • ( K e r f , f K e r f ) , implying ( K e r f , f K e r f ) is a direct summand of (M,A). Since B = f ( g ( B ) ) ~<.f(A) ~
kEZ, nCN},
is an F-homomorphism and for any y ~ Qp/Z:
A(y)=max{A(x)]xE f
i(y)},
so the condition of the Theorem 1.1 o f [7], holds but f is not right invertible ( f is not the fuzzy split in sense of [7]), otherwise, there exists an F-homomorphism
g :(Qp/Z,A) ---+(Qp/Z,A), such that J~/ = l Q,,/z, [ p - i ] denotes the equivalence class of p - i , then f ( g ( [ p - I ] ) ) = [ p - i ] , but . f ( g ( [ p - ~ ] ) ) = p ( g ( [ p - I ] ) ) = g(p[p-~]) = g([0]) = [0], thus [ p - f ] = [0], it is not possible, since p-~ EZ.
J. TanylFuzzy Sets and Systems 105 (1999) 503-507
506
Proposition 3.2. A
morphism f E Hom((M, A), (N,B)) is left invertible if and only if f l m f is a direct summand of (N,B) and A = Bf and f is an R-monomorphism. Proof. If morphism f E Hom((M,A),(N,B)) is left invertible, then there exists a 0 E Hom((N, B), (M,A)) such that 9f = 1M, therefore f is an R-monomorphism. Because A <~Bf<~Agf<<.A, so A = Bf. If y E I m f fq Ker 9, then y = f ( x ) for some x E M and 9(Y) = 0M, thus 0M = 9(Y) = 9(f(x)) = x, SO y = f ( x ) = f ( 0 M ) = 0N, hence I m f fq K e r 9 = {0}. Since every y E N can be written as y = (fg)(Y) + Y - (fg)(Y), where y - (fg)(Y) E Ker9, we deduce that N = I m f ® Ker o. Because A~t<~Bfo<<.Aofo = Ag, so A9 = Bf9. For every Y E N, since
by setting 9(Y)= 9(u + v)= f - l ( u ) , then for all x E M we have
(9f)(x) = g(f(x)) = f - I ( f ( x ) ) = x so that gf = 1M. For every y E N, since
A(O(y)) = A ( f - I ( u ) ) = B(f(f-l(u))) = B(u)
and
B(y) = ( f Im f ) ( u ) A C(v)
= (V{-.,(s)
( f l m f)((fo)(Y)) = V{A(s)ls E M, f(s) =
f(g)(y)}
= B(u)/~ C(v) <~ B(u),
H
I
_
_
,
= V t B ( / ( s ) ) l s EM, f(s)
=
(fg)(y)}
= B((fg)(y)) = A(g(y)) B(y). so
so B <<.Ay, therefore g E Hom((N,B), (M,A)). Since 9 f = Ira, we have f is a left invertible. []
Proposition 3.3. A
morphism f E Hom ((M, A), (N, B)) is invertible if and only if f is an R-isomorphism and A = B f (or f ( A ) = B). It is obvious.
(flmf
+ fKero)(y)
= ( f l m f ) ( ( f o ) ( y ) ) A ( ( f K e r o ) ( y - (fo)(Y)))
Acknowledgements
= B(y) A B(y - (fo)(Y))
I would like to express my thanks for the constructive and helpful comments o f two anonymous referees.
>>.B(y) A B(y) A B((fo)(y))
B(y) and f Im f + f Ker 9 ~
(N,B). If ( I m f , f I m f )
is a direct summand o f (N,B), then there exists an L-fuzzy left R-module (P, C) such that ( N , B ) = ( I m f , f I m f ) @ ( P , C) s o N = I m f ~ P, then for every y E N, y can be written uniquely as y = u + v, such that f ( x ) = u so we can write this element as f - I ( u ) without confusion. N o w define g:N --. M
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