CHAPTER 5

FUZZY PKOJECTIVITY

5.1. Introduction.

Mac Lane [23] formulated projective and injective

lifting properties for the category of abelian groups. Char1es.A .

Weibcl [9] dcscrrbc free and divisible abelian groups

respectlvely; but he did not find the notion projective modules

because he did riot apply these lifting properties to categories of

modules. Cartan and Eilemberg[S] introduced the notion of

projectivlty for modules. The author.studied about fuzzy G-

modules and f u ~ z y representations in [27] and fuzzy injectivity

in [28]. As a continuation of these works here we introduce and

analyse the concept of fuzzy G-module projectivity.

5.2. Fuzzy G-module projectivity.

5.2.1. 1)efinition. k t M and M* be G-modules. Let p and

v be I U L L Y G-n~odules on M and M* respectlvely. Then /r is

v -projective I I

( I) M I \ M* -projective and

5.2.2. Example. If (3 =(~. i , - I , - i \ , M = C and M* = C n , t h e n

M and M* are G-modules ; and M is M*-projectivie ( see

example 2.5.2). Ilcfinc p : M 3[0.1] and v : M * +[0,1] by

11 (x) = 1. i tx =O

_ I ' - h, i f x ( i 0 ) is real

= k , otherwise

v (x) = [ , 11 x =O [ x =(aJ) E M*]

= L, i f a, 1s redl lor all j and a,# 0 for at least one j

= %, otherw~se.

Then p and v are fuzzy G-modules on M and M*

respcctlvcly. Also

Therefore p is v ---proJective , 5.2.3. Proposition. Let M and M* be G-submodules such

that M* is finite dimensional and M is M*-projective. Let

.......... B = ; PI$z , ,P, ) be a basis for M*. Let p and v

be f u z ~ y G-modules on M and M* respectively. If

p ( m j 1 A { v ( p j ) : J = 1 , 2 ,....., n ) forall m ~ M , t h e n is

v-pro~cctive.

I'roof: Let yr r: Hom(M.M*). Then for any m E M , I+I (rn) E M*.

..... So yr (m) = c l Pl+c: Pz+ ..+en Pn , when ci's are scalars, and

hence

v (yr (rn)) == v le, ~ I + C Z Ill+ .......... +c" P")

i .I {v( (3, ) : J =1,2 ,.... n }

1 p (m) , V m E M , by the hypothesis.

81

Thus 1 (m) i 1, ( ' 4 ~ (m)) . V m E M and y, E Hom(M, M*).

Therefore 11 is v --projective , 5.2.4. Proposition. Let M and M* be G-modules and p, v be

fuzzy G-modules on M and M* respectively such that p is

v -projective. I f N is a G-submodule of M* and v1 is any

fuzzy Ci-module or1 N, then p is vl-projective if v / N 2 v I.

Proof: Cr~ven p is v-projective. Therefore (i) M is

M*-projective and (ii) 11 (m) 5 V ( y (m)), for every m & M

and y c Hom(M.M*). Slnce M is M*-projective and N is a

G-submodule of hl*, by proposition 2.5.6, we have M is

N-projective. Now let cp E Hom(M,N) and q : N+ M* be the

inclusion homomorphism. Then 11 *cp = y E Hom(M,M*), and

hence by ( i i ) .

p (m) r v (tl(cp(m))), Vm E M

IL (m) i v (cy(m)), V m E M and cp E Hom(M,N) ( 1 )

Slnce cp (m) t N. we have VI cp(m)) 5 vl( cp(m)) and therefore

by ( 1 1, we get

p (m) 5 v' ( ~ ( m ) ) , Vm E M and cp E Hom(M,N)

Therefore p 1s vl-projective , 5.2.5. Proposition. Let p and v be fuzzy G-modules

on the G-modulcs IM and M* respectively. Let v, (r E [O,l] )

be the fumy G-modules on M* defined as in remark 4.2.7. If

p is v, -projective. for- some r c [0,1] then p is v-projective.

Proof: Assume p is v, -projective, for some r & [0,1]. Then

(11 M 1s M* --pro~ectlve and (11) p (m) 2 v,(yc (m)), V m E M

and y, E Hom(M,M*). Slnce v, L v for all r E [0,1] , and hence

by ( 1 1 ) we have

P (m) 5 v (mj , V m E M and q~ E Hom(M,M*).

Therefore p 1s v --projectivc , 5.2.6. Definition. A fuzzy G-module p on a G-module M is

quasi-projective if p I S F-project~ve.

5.2.7. Example. Let S = { I , I ) and G = Sz = { cp, I+I ),

where q~ =

Let M = span ( 1. I ) over R = C. Then M is a vector space

over K. Def~ne 7': G + GL(M) by x 3 T, where

T, (a+iP) =a X(I) + P.x( i ) ; x E G

Then 1' I S a homomorphism; and therefore M is a G-module.

Also. (he only G-sut)modules of M are ( 0 ) and M itself. Let N

be a G-modulc of M and cp : M + M h ' be a homomorphism.

Co.sc ( i i : I f N = { O / , thcn yr = cp : M + M lifts cp.

Cuse(ii1 : /f N=M. thcn 0 = yr : M + M lifts cp.

Therefore M is quasi-projective. Define p : M + [0,1] by

m = 1 i f m = O

= i f m ; t O

Then I S a t u z ~ y G-module on M and p (m) < p (yr (m)),

for ill1 yr E Hom(M,M). Therefore p is quasl-projective ,

83

3. Further Properties of Fuzzy G-module Projectivity.

5.3.1. Theorem. Let M= (3 M,, where M,'s are G-submodules , = I

of the G-module M. Let p he a fuzzy G-module on M and v,'s

are fuzzy (;-modules on Mi such that v = @ vj. Then p i s i = I

v -projective i f and only i f 11 IS v,-projeetlve, for all i.

Proof: (J) Assume 1 is v -projective. Then ( i ) M is

M=(l) M,-pro~ectlve and (11) p(m) v (\lr(m)), V E Hom(M,M). , = I

To I J ~ { J V : " p is !I,-projective for i=1.2 ....., 11

From ( I ) and from the proposition 2.5.6, we have M is

M, -p r~ jec t i~e for all I . Let cp 6 Hom(M, Mi) and q : Mi -) M

be the ~nelusion homomorphism. Then = r\-cp : M + M is

a homomorphism, and hence by ( i ~ )

Since cp E HomlM, M,), cp(m) E M , and so,

rp (m) = 0+0+.. . ... ...+ cp (m)+.. ...... .+O

. v (cp(m)) = v,(D) A . . . .. , . . A v,( cp (m))A ...... A v,,(O)

= v,( ( ~ ( m )

Theretorc, 11 follows from ( I 1 that

11 (In) c v, (q (m)) , V rn E M and cp E Hom (M,M,).

. . Iherelore p I v,-prc~jectlve for all I

84

( ) Assume C( is v,-projective for all i. Then for each

i =1.2 ..... n, ( a j M is M,-projective and (b) p (m) 5 vi (cp(m)),

for all cp E HomfM, Mi).

To prove t l~ut 11 is v- projective. n

By ( a ) and proposition 2.5.8, M is M= @ Mi -projective. i= l

Let yr E Hom(M, M). Then yr(m) E M, for m E M, and so,

yr (m) = rnl +mz+ .............. +m, . where mi E Mi , V i ( 2 )

Let n, : M 3 M, be the project~on mappings (I i i sn ) . Then

n, ( yr (m)) = m; , for all i and hence by ( 2 ) we get,

yr (m) = n l ( v(m))+ 712( yr(m))+ ...........,. + &(v(m))

Let cp, = n,.yr . Then cp, E. Hom(M, Mi) and therefore

............... ur (m) = cp, (rn)+ cpz (m)+ + cp" (m) ( 3 )

From ( b ). p (m) 5 v ,( cp, (m)) , V m E M and for all i.

..... < A (v, ( cp, (m)) : 1=1,2 n ]

..... From ( 3 ), v (yr(m)) = ;\ ( v , (cp, (m)) : I= 1 2 , ,nl

Hence p(m) i v ( ~ ( m ) ) , V m E M and yr E Hom(M, M).

'Thercl~re );i I S v-projective ,

Corollary. Let M=@ M,, where M,'s are G-submodules of M. , = I

n

Let v,'s be fur .~y G-modules on M, such that v =@ v,. Then ,=I

v 1s quasi-pl.ojective if and only if v 1s v,-projective for all I ,

85

5.3.2. Theorem. Let Mi's be G-modules. Then the direct sum

n

@ M, is quasi-prqjective i f and only i f Mi is Mi-projective, i = J

for every i j E { 1 ,2 ,...., n].

Proof: Follows from the propositions 2.5.6, 2.5.7 and 2.5.8 , Corollary. Let M be a G-module. For a positive integer n,

Mn=M@ M@ .... @ M is quasi-projective if and only if M is

5.3.3. Theorem. lxt M = M ~ O M2, where MI and M2 be

G-submodules of M. Let v,'s are fuzzy G-modules on Mi

(15 J ir) such that v = vl(0 v*. Then v is quasi-projective if

and only i f v, is v,-projective for every i. j & ( 1.2)

Proof (J) Assume that v is quasi-projective. Then (i) M is

M-projective and (ii) v(m) 5 v(y(m)), for all yr E Hom(M,M).

From ( I ) and theorem 5.3.2, M, is M,-projective for i j E { 1,2).

To prove rhur v. is v,-projecrrl~e for every i and j . Since v I S quasi-prcJective, from the corollary of theorem 5.3.1,

v is v , -projective for i=1,2. ( 1 )

. . ~ ( r n ) : v, ( cp(m)), V q E Horn(M, M,), i = I ,2. ( 2 )

Let yr E Horn(M1.M~) and let rrJ: M 3 MI be the projection

map. Then ty-nl : M + M2 IS a homomorphism. By ( 2 ),

we get

I f rn = rn, EM,, then

v(mli <; vz ( ( w - ~ I (mi))= ~2 (v(m1)).

Also.

v(ml) = v (ml+0)

= V I (ml) A vz(0)

= V I ( m ~ ) A V I (0)

= V I (ml)

vl(ml) 5 v2 (v(ml)), V y, E Hom(Ml,M2) and ml E MI

Hence vl is vz.-projective. Similarly we can show that v2 is vl-

projcctlve.

Now to prove 1.1 i.s vl-projective. Let 0, E Hom(M1,MI). Then

el*nl E Hom(M, MI) . Since v is v-projective, by ( 1 ), v is

vl-projective.

v (m) < V I ( O I - Z I (m)). V H I 6 Hom(MI,MI) and m E M .

If m=ml E M ) ,

v (ml) : V I ( O I - ~ I (mi)) = V I ((01 (ml))

But, we have, v(ml)= vl (mi) . and so

Therefol-e vl IS v1-pro~ective. Similarly we can show that v2 is

v2-projcctlve. Hence we get v, is v,-projective for every

l , J f ; [ 1 , 2 ] .

( Assume thal v, 1s v,-projective for any i, J E {l,2].

Then (111) M, 1s "9-projective for every i , j E 11.2) and

( I ) v r n j 5 v, i ) , for all y, E Hom(Mi, M,). By (iii) and

87

lheorern 5.3.2, M=MI@ M M ~ is M=M~O Mpprojective, and

hencc by proposition 2.5.8. M is M,-projective for i =1,2.

First ro prove tlraf v is v,-prujective. Let 81 E Hom(M, MI)

and let cpl : Mz 3 M be the inclusion homomorphism. Then

01.cp1 M2 3 MI is a homomorphism. Since v~ is vl-projective,

v:(mz) 5 V I (01.cp1 (md), v m2 E M2 and 81 E Hom(M, MI)

Slnce 01 E Hon>(M,Ml), ql = 011 M I F Hom(Ml,Ml). Also, since

vl 1s vl -projective. we have V I (ml) < VI (111 (ml)) .Therefore

v l ( m l ) 5 v ~ i 0,iml)). v m~ t: MI and 81 E Hom(M,MI) ( 4 )

From ( 3 ) and ( 4 ), we get, V mi E Mi and 01 E Hom(M, M I )

V I (ml ) A v2 (mz) i- V I ( 81 (ml)) A V I (81 (m2))

If m=rnl+mz E M=M~O M2, then v (m) = vl (ml) A v2 (m2),

and hence

v (m) 5 vl ( 01 cml)) A vl (01 (md) ; 81 E Hom(M, MI) ( 5 )

Since v , is a fuzzy G-module on MI , we have, V x, y E MI

and scalars J, h

vl (ax+by) 2 vl(x) A vl(y) ( 6 )

Slncc (II E Hom(M,Ml), x = 01 ( m ~ ) E MI and y = @ I (m2) E MI.

Therefore, bq ( 6 )

vl ( lx+ly)= vl (1.01 (m1)+1.01 (mz))

2 V I ( 0 1 ( m ~ ) ) A V I (01 (m2))

* V I (01 (m)) -' V I ( 01 (ml))A V I (01 (md)

From ( 5 ) and ( 7 1,

v (rn) 5 vl (0, (m)), for all 81 E Hom(M,M))

Therefore v is vl -projective. Similarly we can show that v is

v2-projective. Therefore v I S v, -projective for each i =1,2.

Hence, by corollary of 5.3.1, v is quasi-projective ,

5.3.4. Theorem. Let. M =a Mi, wher Mi's are G-submodules ,=I

of the G-module M. Let v,'s be fuzzy G-modules on M, n

(15 I 5n) such that \I = @I V, Then v 1s quasl-project~ve ~f and ,=I

only i f v, 1s v,-projective for I , J E (1.2 ,......., n) .

Proof: Follows from the preced~ng theorem . rn

5.3.5. Theorem. Let M=@ M, and M * = O N,, be G-modules, , = I ,= I

where Mi's ant1 N,'s are G-submodules of M and M*

respect~vcly. Thcn both M and M* are relatively projective

and relarlvely lnjcctlve. If p and v are fuzzy G-modules on

M and M* rcspectrvely, then p is v -injective if and only

if v I S p projective.

Proof: First t o p , r v M is M *-injective.

Let X be any G-submodulc of M* and cp : X 3 M* be a

homomorphism. Then, we have, three cases : (i) X = (0).

k

(ii) X = N,, fol.somej (1:; 1 Sm) and (iii) X = @ N,, (k Sn). j=I

COSP /i) l f X = /Of : then the zero homomorphism

O=!y : M* + M extends cp .

Cuse (ii) IJ X = N,, for .sortre j (134n) : then the mapping

yr : M* + M defined by

tq (m*)= 9 (n,), where m* =n,+nz+..+n,+ .... +n, E M*

is a homomorph~srn and y extends cp.

C u e ( I ) I j X = (0 N , ( i j: then the homomorphism ,,=I

yr(m*) = cp(nl+nz+ .... +nk), where m* = nl+nz+ ...+ nt+ ...+ n, E M*

extends c p .

Thus in all cases, the homomorphism cp: X 3 M can be

extended to a homomorph~sm I+I : M* + M; and so M is

M*-injective. Similarly we can show that M* is M-injective.

Hence M and M* are relatively injective.

Now 10 prove M is M*-pr((j(>etivc.

Let N* be any G-submodule of M* and cp : M + M*l N* be

a homomorph~srn. Then, as above we have three cases.

Casc ( I ) If A'* = /O / : then the homomorphism

cp : M + M*fN* - hl* itself l~ f t s cp .

Case ( i i ) l f N* = N, , Jor sorrre.j (1 5 j 5 111) : then let

cp : M + M*l N = N,@ Nr ........ @ N,.I@ N,+I ........ @ Nm

be the glvcn homomorphism. Define yr : M + M* by

yr (m)= m* ,where cp (m) =m* +N*

90

Then yr is a homtrmorphlsm. Further, if A : M* + M*/N* is

the projection map, then for any m E M,

(n-v )(m) = ~ ( y r (m)

= n(m*) ,where cp (m) = m* + N*

= ni*+N*

Therefore n*yr .; cp, and so yr lifts cp.

k

Case (iii) IS N* = @ N, . (' k 2' 11) : then let ,=I

cp: M 3 M*/N* = Nt+l(D ............ @ N, be the given

homomorphism Then as in case (ii), the mapping y~ : M + M*

defined by

yr (m)=: m* , where cp (m) =m* +N*

is a homomorphrsm which hfts cp.

Therefore in all the cases, any homomorphism from M into

M*IN* can be llfted to a homomorphism from M into M* and

hence M is M*-projective. Similarly we can show that M* is

M-projective. Hcnce M and M* are relatively projective.

Proof of the .sc?corzd part:

(4 ) Assumc )I IS v -rnlccllve Then

v(m*) 5 p(q(mt)) , V 11 E Hom(M*,M) and m * ~ M*

Since M and M* are relatively projective, we have M* is

M-projective. Hence v is )i-projective.

(t) Assume v is p-projectrve. Then

v(m*) L p(q(m*)) . '-J rl E Hom(M*,M) and m * ~ M*.

9 1

S~nce M and M* arc rclativcly injective, we have M is

M*-injective and so p 1s v -injective

5.3.6. Remark. It follows from the above theorem that any two

finite dimensional G-modules are relatively injective and

relatively projective. Further, considering a pair of fuzzy

G-modules on such two G-modules, we get, a dual relationship

betwcen their injectivity and projectivity I

n

Corollary (1). Let M = ul M,. where M,'s are G-submodules I= I

Then M 1s quasi-injective and quasi-projective. If p is any

fuzzy G-module on M. Then is quasi-injective if and only if p

Proof: Follows from the preceding theorem [ put M* = M and

v = P I m

n

5.3.7. Remark. Let M =(i) M, , where M,'s are G-submodules , = I

of M. Then from theorem 5.3.5, we have M is both quasi-

lnjectlvc and quasi-projectlvc. S ~ n c e M 1s quasl-lnjectlve, by the

corollary of theorcm 4.3.3. M, is M,-injective for all i and j.

Also, slnce M I S quasi-prc)jectivc. by theorem 5.3.2, Mi is

MI-projcct~vc tor all I and J Hence for for each I, M, 1s both

quasi-ln~cctive and quasi-projective.

Let v = a V, where v,'s are fuzzy G-modules ,=I

92

on M,. Then, from theorcrn 5.3.4, we have v is quasi-projective

i f andonly i f v, is v,-projective for i , j E (1 .2 ......, n).Hence

v I S quasi-projective i f and only if vi is quasi-projective for

i = 1.2, ...,n . Corollary (2) . k t M be a finite dimensional G-module of

dimens~on at least 2. Then there exists infinite number of fuzzy

completely reducible G-modules v, ( r E (0,1] ) such that for

the fuzzy completely reducible G-module v in the theorem

3.3.6 satisfies v is v,-injective and v, is v -projective.

Proof: Since M is finite dimensional, from theorem 5.3.5, we

have M is both quasi-injective and qausi-projective. In the

theorem 3.3.6, we have the level subset of fuzzy completely

Choose r E ( O , I j such that r 5 II~+I Then from the propositioin

3.3. LO, the fuzzy completely reducible v, satisfies

v,(m) 5 A {v (aj) : j =1,2 ,..., n)

where (ul,a2 ,......., a,] is a basis for M. Hence by proposition

4.2.4, v is v, -injective and by proposition 5.2.3, v, is

v -prqjective , 5.3.10. Kemark. Correspond~ng to any proper decomposition

M=M,(& MI, the fumy completely reduc~ble G-modules

v and v, in the preceding corollary has decomposition ,~. v = v,l!? vz and v, = v ~ ~ ' , ! ; ~ v ~ , . where vl, vl, are fuzzy

93 G-modules o n MI and where v2, v2, are fuzzy G-modules on

M2 repectlvely.

Also i t follows from the from the preceding

proposition that v is v~~njec t ive and v, is v-projective.

Further by theorems 4.3.2 and 5.3.1, v is vi,-injective and

v, 1s v,-projective for all i = I - 2 , Corollary (3). Any finite d~mensional G-module has a

fuzzy G-module which 1s both quasi-injective and quasi-

projective.

Proof: Let M be a finite dimensional G-module . Then by

proposit~on 2.4.12, we have M = MI@ M*@ ....... @ Mn . where

M,'s are G-submodules of M

Slnce M = :I) M, , from theorem 5.3.5, we have M 1s both ,=I

quasl-~njectlve and quasi-projective, Define q : M +[O,l] by

I ) 1 ; ~f m=O ( 1 )

=: t ; I S m f 0 and where t E [0,1] is fixed.

Then 11 1s a fuzzy G-module on M.

Now I L . ~ w'ill prove 11 is quusi-i~~jecfive

Let v r: Hom(M.M). Then for any m E M , yr(m) E M.

(i) /fr11=0, then y(m)=O. Hence by ( 1 )

q(mI= q(y(m)) = 1

(ii) If r n f 0 urld y/(nl)=O, then

q (m)= t and q(v(m)) = q ( 0 ) = I

Therefore

!f w f r r l i go , then q(m)= q ( ~ ( m ) ) = t ( 4 )

From ( 2 ). ( 3 1 and i 4 1. q(m) q(v(m)), b' I+I E Hom(M, M).

Therefore r1 IS quasi-projective. Also by theorem.5.3.5, q is

quasi-lnjectlve. Herice the result , 5.3.11. Example. Let G = ( , xp) and let F = (&,+p, xp) be

the field. Lel M .- ~ ( 4 2 ) = I a+b42/ a,b, E F ) . Then M is a

vector space over F. For g c G and m= a+b& M, define

g.m = g.( a+bd2) = (g xp a) + (g xp b) 42

Then gm c M and satisfies

( I ) g.(ml+mz) = gml+gmz

(11) (g g L j m = g(gl(m))

(111) 1.m = rn, for all m,ml,m2 E M and g, g ' ~ G 2 Therefore M 1s a G-module. Let M* = F = ( (a,b) / a,b E F).

For g L: G ,m* = (a, 6 ) c M* , define

g.mt = g.(a,b) = (g xp a, g xp b) E M*

Then M* is also a G-modulc.

Further. M = MI?) M2, where M I = F and M2 = 4 2 ~ are G-

submodulcs of M and M*= NI@ N2, where Ni =&IF and

N1= *:2F arc G-submodules of M* and E I = (1.0). E Z = (0,l).

Hence as in the proof of the theorem 5.3.5, we can show that M

and M* are rcla~~vcly projective and relatively injective. Now

del'ine p . M 3 [O,l J and v : M* + [O, I] by

= i f r n t O

Then p and v are fuzzy G-module on M and M* respectively.

Also

v(rn) < p ( l ) A p(J2), forall EM*.

Then from proposition 5.2.3, v is p-projective, and by

theorem 5.3.5, p 1s v-~njectlve , 5.3.13. Example. Let G = { 1,-1 ) and let M = Cn (n _> 1).

Then M 1s a 2n-dimens~onal G-module over R. Let M* = Rm

(m 21). Then M* IS also a G-module.

Cons~der the basis ( El ,&: .........., c2I ,...... E ~ ' ) for M over

R, where

r . , = (0.0 ...., 1 ,...., 0,0) [ I is in the i ' place] and

i:j ' = (0.0 ,..... i ,.... 0.0) [ i = J - 1 is in the jIh place].

Lct M , = E , K ( l < i < n ) and M , + , = E ~ ' R ( I < j < n ) . Then

M,'s ( 1s I <n ) and M,+,'s ( I Sj 5n) are G-submodules of M Ln

and M =iD M, , Let { al,a2 ,......, a,) be the standard basis for ,=I

M* over K. Then M* =@ Nk, where Nk=atR ( I < k ilm) are k = l

96

G-submodules of M*. Here M and M* are the direct sums of

finite number of its G-submodules and hence by theorem 5.3.5,

M and M* are relatively injective and relatively

prolectlve. Now define p : M 3 [0,1] and v : M* + [O,l]

by

............... . p m = I , i f m=O where m=(xl.x2 9x1") & M.

:= %, if mf 0 and all x,'s are real

= 11, otherwise.

Then p and v are fuzzy G-modules on M and M* respectively.

Also,

.... ..... v (x ) 5. { A (p (61 11 : i=1.2 n)) A { A ( p (cj ' ) : ~ ~ 1 . 2 n)),

for all x E M*. Hence, by proposition 5.2.3, v is p-projective

and from theorem 5.3.5, p is v -injective , 5.3.14. Proposition. For any pnme p and any integer n>O, there

exists a G-module M of order p" which is both quasi-injective

and quasi projcct~ve. Further. there exists a quasi-injective and

quasl-pruject~ve fuzzy G-module on M.

Proof: Let M be the field having p" elements. Then M has

a subfield K, which 1s isomorphic to 2,. Then M is an n-

dirncns~onal vector space over K. Let G = K-10). then G is a

mult~pl~cative subgroup of M and so M is a G-module. Let

...... 1 a ! , al , . a,,] be thc basis for M. Then

97

M = MIo:j M L O ....... (53 Mn ; where M, = ajK (l<j<n) are n

G-submodules of M S ~ n c e M = @ M, , by corollary (I) and ,=I

corollary (3) of 5.3.5, we have M is both quasi-injective.

quasl- projective and M has a fuzzy G-module, which is both

quasl-injective and quasi-projective , 5.3.15. Proposition. If p 1s a prime and n is a composite

pos~tive integer, then there exists a finite group G and finite G-

modules M and M* such that they are relatively injective and

relat~vely projective. Further, there exists fuzzy completely

reduc~ble G-modules on M and M* respectively.

Proof: Let M be a field hav~ng pn elements. Since n is not a

prime, there exists a proper subfield M* of M having pm

elements, where m is a divisor of n. Also M and M* have a

subfield K which is isomorphic to Zp. Let G = K-{O). Then

M and M* are 'n' and 'm' dimensional G-modules over K.

Then we can choose basis {a1,a2 ,....... a,) for M* SO that

{ a 1 a . . am,am+l,.... a ) i s a basis for M (m <n); and hence

M* =: K ~ I R ! :!a2@. . . . . @ Ka, and

M =Ka1@ Ka2O. . a Ka,@ Ka,+l@ .... (9 Ka...

Then by proposition 5.3.5. M and M* are relatively injective and

relat~vcly prolective and hy proposition 3.3.6, M has a fuzzy

completely reduc~ble G-module v on M. Then vl M* i:; a fuzzy

completely reducible G-module on M*,

98 5.3.16. Example. Let p=3 and n=6. Let M={zeroes of the

polynomial xlZ9--x over 2 3 ) and M* ={zeroes of the

polynomial xZ7-x over %,I. Then M and M* are fields of

characteristic 3. Then K=Z, is a subfield of M, M* and M* is

a subfield of M. Let G=Z3-(0). Then both M and M* are G-

modules over K. Let MI=M-{OJ and Mz= M*-(0). Then by

remark 2.6.2, we have M I and Mz are cyclic groups. Let MI=

( a ). Then M2= ( ), where $=ak where k > l . We have, M=K(a)

and M*=K(P). Also, M is a 6-dimensional vector space and M*

is a 3-dimensional vector spaces over K. Since M* is a subfield

of M, there exists a bas~s {a1,a2,a3) for M* so that

{a , ,a2 ,...., & ) is a basis for M and hence

M* =: Kal(!j K a z B Ka3 and

Then the set v defined in the theorem 3.3.6 is a fuzzy

completely reducible G-module v on M and V / M. is a fuzzy

complerely reducible G-module on M* , 5.3.17. Proposition. Some completely reducible G-modules

havc tuzzy conlplerely reducible G-modules which are both

quasl-~njective and quasi-pro~cctive.

Proof: Let [I be a prime and n 22 be an integer. Let M be the

field having p" elements. Then M has a subfield K, which is

isomorphic to ZI,. Let G = I(-(01. Then M is a G-module and is

an n-dimensional vector space over K. Since M is finite

dlrnenslonal, by ihe corollary of 2.3.9, i t 1s completely reducible

and since 1122, by proposltlon 5.3.5, M is quasi-injective and

quasi-projective. l x t ( a1 . a~ ,..... a,) be a basis for M over K n

and v =@ v, be the corresponding fuzzy completely reducible i= 1

G-module obtalned by theorem 3.3.6. Since M = Zpn, it is

cyclic. Let M =: ( a ) and q c Hom(M,M). Then cp is determined

by the value of the homomorphism on the generator 'a'.

Therefore elther cp =O or an isomorphism. Define p: M+ [0,1]

by cl (m) = A (v jyr (m)) : yr E Hom(M, M))

Then )* 1s a furz,y G-module. For let x, y E M ; a, b E K and g E G,

i' p(x) '4 P(Y)

(ii) p(gx) = A ( V (yr(gx)) . y E Hom(M. M))

= 11 ( v 1.g yr(x)) : y E Hom(M, M)]

1. '1 ( v ( ~ ( x ) ) : y E Hom(M, M))= p(x)

We have M = M,(D M2ji) ......... @ M., where Mi =a,K are

G-submodules of M. For each i. the map & :Mi + [O,l]

defincd by

~ ( m ) = '2 I v, (y, (m)) : vi E Hom(M,, Mi))

are f u ~ z y G-modules on M, and p = @ p, Therefore p is a ,=I

completely reducible G-module.

Now we will pruve p is quasi-injective

Let 41 E Hom(M, M) and m F, M. Then we have

ii(cp (m)) = A I V(W fcp(m)) : w E Hom(M, MI1

= A ( v(v *cp(m)) : y~ E Hom(M, M))

2 11 {v (W (m)) : y~ E Hom(M, M)) = p(m)

~ ( m ) 2 p( cp(m)), V cp E Hom(M,M) and rn E M

Therefore p I S quasl-injective and by corollary (1) of 5.3.5,

p I S quasr-projective .