PAUL ISAAC
Received 30 June 2004 and in revised form 18 November 2004
The concepts of free modules, projective modules, injective modules, and the like form an important area in module theory. The notion of free fuzzy modules was introduced by Muganda as an extension of free modules in the fuzzy context. Zahedi and Ameri introduced the concept of projective and injectiveL-modules. In this paper, we give an alternate definition for injectiveL-modules and prove that a direct sum ofL-modules is injective if and only if eachL-module in the sum is injective. Also we prove that ifJis an injective module andµis an injectiveL-submodule ofJ, and if 0→µ−→f ν−→g η→0 is a short exact sequence ofL-modules, thenνµ⊕η.
1. Introduction
Though the notion of a fuzzy set was introduced by L. A. Zadeh in 1965, its application to algebraic concepts started only in 1971 when A. Rosenfeld introduced fuzzy subgroups of a group. Tremendous and rapid growth of fuzzy algebraic concepts resulted in a vast literature. The book of Mordeson and Malik [7] gives an account of all these up to 1998. The notion ofL-modules as an extension of classical module theory is available in this book. However, there are many concepts in abstract algebra which are to be analyzed in the fuzzy context. The notion of free fuzzy modules was introduced by Muganda [8] as an extension of free modules in the fuzzy context. The concept of a freeL-module is available in [7]. Zahedi and Ameri [9] introduced the concepts of fuzzy projective and injective modules.
In our earlier paper [5], we introduced an alternate definition for a projective L-module and proved some related results. In this paper, inSection 2, we give the essential preliminaries and inSection 3, we give an alternate definition for an injectiveL-module and prove some results using this definition. Throughout this paper, unless otherwise stated,L(∨,∧, 1, 0) represents a complete Brouwerian lattice with maximal element “1” and minimal element “0;”Ra ring with unity “1” andMa left module overR. “∨” de-notes the supremum and “∧” the infimum inL. We callLa regular lattice ifa∧b >0 for alla,b >0 inL. “⊆” denotes the inclusion and “⊂” the proper inclusion. The set of all L-subsets ofM, that is, the set of all functions fromMtoL, is denoted byLM.
Copyright©2005 Hindawi Publishing Corporation
Forx∈M,a∈L, theL-subset which takes the valueaatxand 0 elsewhere is denoted bya{x}. That is,
a{x}(y)=
a0 ififyy= =x,x. (1.1)
2. Preliminaries
In this section, we review some definitions and results which will be used later. For details, reference may be made to Mordeson and Malik [7], for preliminaries regarding lattices Birkhoff[1], and for theory of modules, Goodearl and Warfield [2] and Hungerford [3]. Definition 2.1(see [7]). Forµ,ν∈LM, defineµ+νand−µas follows.
Forx∈M,
(µ+ν)(x)= ∨µ(y)∧ν(z) :y,z∈M,y+z=x, (−µ)(x)=µ(−x). (2.1)
Also for an arbitrary familyµi∈LM,i∈I, ofL-subsets ofM, define
i∈Iµi(x)= ∨
∧
i∈Iµi
xi :xi∈M,i∈I, i∈Ixi=x
, (2.2)
where in the expressionx=i∈Ixi, at most finitely manyxi’s are not equal to 0. Definition 2.2(see [7]). Forµ∈LM, define the following:
(i)µ∗ = {x∈M:µ(x)>0}, called the support ofµ,
(ii) fora∈L,µa= {x∈M:µ(x)≥a}, called thea-cut ora-level subset of µ, and µ>
a= {x∈M:µ(x)> a}, called the stricta-cut or stricta-level subset ofµ. Definition 2.3(see [7]). Let f be a mapping fromXintoY, and letµ∈LX andν∈LY. TheL-subsets f(µ)∈LYand f−1(ν)∈LX, defined by, for ally∈Y,
f(µ)(y)= ∨
µ(x) :x∈X, f(x)=y iff−1(y) =φ,
0 otherwise, (2.3)
and for allx∈X,
f−1(ν)(x)=νf(x) , (2.4)
are called, respectively, the image ofµunder f and the preimage ofνunderf. Definition 2.4(see [7]). Letµ∈LM. Thenµis said to be anL-submodule ofMif
(i)µ(0)=1,
Sayingµis a leftL-module means thatµis anL-submodule of some left moduleM over a ringR. The set of allL-submodules ofMis denoted byL(M).
Remark 2.5. We note from [7] that ifµ,η∈L(M), thenµ+η∈L(M). Also ifµi∈L(M), i∈I, theni∈Iµi∈L(M). From [6], we see thatµ∈L(M) if and only ifµais anR-module for alla∈L.
Definition 2.6(see [7]). LetMandNbeR-modules and letµ∈L(M) andν∈L(N). An isomorphism f ofM ontoN is called a weak isomorphism ofµintoνif f(µ)⊆ν. If f is a weak isomorphism ofµintoν, then say thatµis weakly isomorphic toνand write µ ν.
An isomorphismf ofMontoNis called an isomorphism ofµontoνif f(µ)=ν. If f is an isomorphism ofµontoν, then say thatµis isomorphic toνand writeµ∼=ν. Definition 2.7 (see [4]). Let Ai,i∈Z, beR-modules and letµi∈L(Ai). Suppose that
··· fi−1
−−→Ai−1
fi
−→Ai−−→fi+1 Ai+1
fi+2
−−→ ··· is an exact sequence of R-modules. Then the se-quence ··· fi−1
−−→µi−1
fi
−→µi−−→fi+1 µi+1
fi+2
−−→ ··· of L-modules is said to be exact if, for all i∈Z, the set of integers,
(i) fi+1(µi)⊆µi+1,
(ii) fi(µi−1)(x)>0 ifx∈Kerfi+1and fi(µi−1)(x)=0 ifx /∈Kerfi+1.
Definition 2.8(see [4]). Let 0→A−→f B−→g C→0 be a short exact sequence ofR-modules. Letµ∈L(A),η∈L(B), andν∈L(C). Then an exact sequence ofL-modules of the form 0→µ−→f η−→g ν→0 is called a short exact sequence ofL-modules.
Definition 2.9(see [4]). Let
0 A f
φ
B g ψ
C ξ
0
0 A f
B g C 0
(2.5)
be two isomorphic short exact sequences ofR-modules with the given isomorphismsφ, ψ, andξ. Letµ∈L(A),ν∈L(B),η∈L(C),µ∈L(A),ν∈L(B), andη∈L(C) be such that
0 µ f ν g η 0, (2.6)
0 µ f
ν g η 0 (2.7)
are short exact sequences ofL-modules. Then the sequence (2.6) is said to be weakly isomorphic to the sequence (2.7) ifϕ(µ)⊆µ,ψ(ν)⊆ν, andξ(η)⊆η.
Definition 2.10(see [7]). Letµ,η,ν∈L(M). Thenµis said to be the direct sum ofηand νif
(i)µ=η+ν, (ii)η∩ν=1{0}.
In this case, writeµ=η⊕ν.
Definition 2.11(see [4]). LetAandBbe twoR-modules,µ∈L(A),η∈L(B). Consider the direct sumA⊕B. Extend the definition ofµandηtoA⊕Bto getµandηinL(A⊕B) as follows:
µ(x)=
µ(x)0 ififxx /∈∈A,A, i.e.,µ(a,b)=
µ(a)0 ififbb= =0,0, for (a,b)∈A⊕B,
η(x)= η(x)
ifx∈B,
0 ifx /∈B, i.e.,η (a,b)=
η(b)
ifa=0,
0 ifa =0, for (a,b)∈A⊕B. (2.8)
Thenµ,η∈L(A⊕B). Moreover
µ∩η (x)=µ(x)∧η(x)=
1 ifx=0,
0 ifx =0. (2.9)
Thereforeµ+ηis in fact a direct sum and is denoted byµ⊕η. Remark 2.12. Note that
(µ⊕η)(a,b)
=(µ+η)(a,b)
= ∨µa
1,b1 ∧η
a2,b2 :
a1,b1 ,
a2,b2 ∈A⊕B;
a1,b1 +
a2,b2 =(a,b)
=µ(a, 0)∧η(0,b)=µ(a)∧η(b).
(2.10)
Definition 2.13(see [7]). Letµ,µi∈L(M), for alli∈I, thenµis said to be the direct sum of{µi:i∈I}, denoted byi∈Iµi, if
(i)µ=i∈Iµi,
(ii)µj∩i∈I−{j}µi=1{0}for allj∈I.
3. InjectiveL-modules
if each summand in the sum is injective. Also we prove that ifµ∈L(J) is an injective L-module, and if 0→µ−→f ν→−g η→0 is a short exact sequence ofL-modules, thenνµ⊕η. Definition 3.1. LetJ be an injectiveR-module and letµ∈L(J). Thenµis said to be an injectiveL-module if forR-modulesA,B, andη∈L(A),ν∈L(B),gany monomorphism fromAtoBsuch thatg(η)=νong(A), and f :A→J anyR-module homomorphism such that f(η)=µon f(A), there exists anR-module homomorphismh:B→J such thathg= f andh(ν)⊆µ.
From the crisp module theory, it is known that anR-moduleJis injective if and only if every short exact sequence 0→J−→f B→−g C→0 splits so thatB∼=J⊕C. An analogous result exists in the case ofL-modules.
To prove this we need the following theorem.
Theorem3.2 (see [4]). Let0→A1
f
−→B−→g A2→0be a short exact sequence ofR-modules
and letµ1∈L(A1),µ2∈L(A2),η∈L(B)be such that 0→µ1
f
−→η−→g µ2→0 is a short
exact sequence ofL-modules. If there exists anR-module homomorphismk:B→A1with
k f =IA1, the identity map onA1, such thatk(η)⊆µ1, then the given short exact sequence
0→µ1
f
−→η−→g µ2→0is weakly isomorphic to the short exact sequence0→µ1−→i µ1⊕µ2−→π
µ2→0. In particularηµ1⊕µ2.
Theorem3.3. LetJbe an injective module andµ∈L(J)an injectiveL-module. If0→J−→f
B−→g C→0is a short exact sequence ofR-modules andν∈L(B)andη∈L(C)are such that 0→µ−→f ν−→g η→0is a short exact sequence ofL-modules where f(µ)=νonf(J), thenνis weakly isomorphic toµ⊕η. That is,νµ⊕η.
Proof. SinceJ is injective, it is well known that any short exact sequence 0→J−→f B−→g
C→0 splits andB∼=J⊕C, and the sequence 0→J−→f B−→g C→0 is isomorphic to 0→
J−→i J⊕C−→π C→0. Now sinceµ∈L(J) is an injectiveL-module, and since f(µ)=νon f(J), from the definition, we geth(ν)⊆µ. Thus there exists a homomorphismh:B→J such thath f =IJ andh(ν)⊆µ. Then, by the above theorem, 0→µ−→f ν−→g η→0 is weakly isomorphic to 0→µ−→i µ⊕η−→π η→0, in particularνµ⊕η. In the crisp theory, we have the theorem that a direct sum of modules is injective if and only if each summand is injective. The same is true in the fuzzy case.
Theorem3.4. LetQα,α∈Ibe injectiveR-modules andµα∈L(Qα),α∈I. Thenα∈Iµα∈ L(α∈IQα)is injective if and only ifµαis injective for allα∈I.
Proof. As we have already mentioned, the moduleα∈IQαis injective if and only ifQα is injective for allα∈I. Letiα:Qα→α∈IQα andπα:α∈IQα→Qαbe, respectively, the canonical injection and projection. Obviouslyα∈Iµα∈L(α∈IQα) and suppose
fα(η)=µαonfα(A), then we have to show that there exists anR-module homomorphism hα:B→Qαsuch thathαg=fαandhα(ν)⊆µα,
0 A g
fα B
hα
Qα
iα
Qα
πα
(3.1)
Let α∈IQα be injective. Consider iαfα:A→α∈IQα. First of all, we show that (iαfα)(η)=α∈Iµαon (iαfα)(A).
We have (iαfα)(η)∈L(α∈IQα), and ifx=xα∈(iαfα)(A)⊆α∈IQαwherexα∈ Qα(α∈I), thenx=(iαfα)(a) for somea∈A. That is,x=iα(fα(a)) where fα(a)∈Qα. Then
iαfα (η)(x)= ∨η(a) :a∈A;iαfα (a)=x
= ∨η(a) :a∈A;iαfα (a)=iαfα (a)
= ∨η(a) :a∈A;iαfα(a) =iαfα(a)
= ∨η(a) :a∈A; fα(a)= fα(a).
(3.2)
Also
α∈I
µα(x)= ∨
∧µαxα :xα∈Qα,α∈I;x= α∈I
xα
=µαfα(a) since the supremum is attained for the direct sum decompositionx=0 + 0 +···+ 0 +fα(a) + 0 +···+ 0
= fα(η)fα(a) = ∨η(a) :a∈A; fα(a)= fα(a).
(3.3)
From (3.2) and (3.3), we get (iαfα)(η)(x)=α∈Iµα(x) for allx∈iαfα)(A).
Now sinceα∈Iµαis injective, we get thatiαfα:A→α∈IQαhas an extensionk:B→
α∈IQαsatisfyingkg=iαfαandk(ν)⊆α∈Iµα. Takehα=παk. Thenhα:B→Qαis an extension offα:A→Qαsatisfyinghαg=fα. It remains to prove thathα(ν)⊆µα.
We havek(ν)⊆α∈Iµα. Therefore
παk(ν) ⊆πα
α∈I µα
Now forxα∈Qα(α∈I),
πα
α∈I µα
xα
= ∨
α∈Iµα(y) :y∈
α∈IQα;πα(y)=xα
=µαxα since the supremum is attained fory=0,..., 0,xα, 0,..., 0 . (3.5)
Thusπα(α∈Iµα)=µα, and so from (3.4), we get (παk)(ν)⊆µα. That is,hα(ν)⊆µαas required.
Conversely, suppose thatµαis injective for allα∈I. We prove thatα∈Iµαis injective,
0 A
f
παf g
B
kα
k Qα
iα
Qα
πα
(3.6)
SinceQαis injective for allα∈I, we have thatα∈IQαis injective. LetA,BbeR-modules, η∈L(A),ν∈L(B),gany monomorphism fromAtoBsuch thatg(η)=νong(A), and suppose that f :A→α∈IQα is a module homomorphism satisfying f(η)=α∈Iµα on f(A). Since f(η)=α∈Iµαon f(A), we get that (παf)(η)=πα(α∈Iµα)=µα on (παf)(A). This together with the fact that eachµαis injective imply that eachπαf :A→
Qα admits an extension kα:B→Qα such that παf =kαg. These homomorphismskα givek:B→α∈IQαsuch that παk=kα and for eachx∈A, (παk)(g(x))=kα(g(x))= (παf)(x) for allα∈I. Thereforek(g(x))=f(x) for allx∈A. Thereforekis an extension of f such thatkg=f and alsokα(ν)⊆µα. Now
kα(ν)⊆µα=⇒παk (ν)⊆µα=⇒παk(ν) ⊆µα=πα α∈I
µα. (3.7)
Since this is true for everyα∈I, it follows thatk(ν)⊆α∈Iµα. This completes the proof.
Acknowledgments
References
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[2] K. R. Goodearl and R. B. Warfield Jr.,An Introduction to Noncommutative Noetherian Rings, London Mathematical Society Student Texts, vol. 16, Cambridge University Press, Cam-bridge, 1989.
[3] T. W. Hungerford,Algebra, Graduate Texts in Mathematics, vol. 73, Springer, New York, 1980. [4] P. Isaac,Exact sequences ofL-modules, submitted.
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[8] G. C. Muganda,Free fuzzy modules and their bases, Inform. Sci.72(1993), no. 1-2, 65–82. [9] M. M. Zahedi and A. Ameri,On fuzzy projective and injective modules, J. Fuzzy Math.3(1995),
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Paul Isaac: Department of Mathematics, Bharata Mata College, Thrikkakara, Kochi 682 021, Kerala, India