near rings

8/19/2019 Near Rings

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41

CHAPTER III

BOOLEAN NEAR RINGS AND WEAK COMMUTATIVITY

3.1 Boolean Near-Rings

Definition: 3.1.1

A Near –Ring R in whih e!er" e#e$en% i& an

i'e$()%en% i& a##e' B))#ean Near Ring*

+i*e*, -. / - 0 1)r eah - ϵ N

Weak Commutativity

Lemma:3.1.2* I1 N i& a 2))#ean near3ring 0 %hen -" / -"- 0 1)r eah -0" ϵ N*

Proof:

-" / -." +&ine0 - ϵ N,

/- - " +&ine0 N i& )$$4%a%i!e,

/- " - +&ine0 E!er" B))#ean Righ% Near

Ring R i& wea5#" C)$$4%a%i!e,

There1)re0 -" / -"- 0 1)r eah -0 " ϵ N

!eorem 3.1.3. I1 N i& a 2))#ean near3ring0 %hen a2 / a2 1)r eah a020 ϵ N*

Proof. Le% a020 ϵ N*

Then 2+a 3 a, " 2+a # a.,

" 2+a # a,

" 2+6,*

Th4&0 7+a3a,2+a3a,8 / +a3a,2+6,



42

/9 +a3a,2 / +a3a,2+6, +2" Le$$a :*;*.,

/9 a23a2 / a2+6, – a2+6, +Sine 2 / 2,<<E=n*;

Ne-%0 +a 3 a, / +a 3 a,

/ +6,

/9 +a 3 a, / +6, + 2" Le$$a :*;*. ,

/9 2+a 3 a, / 2+6,

Again0 2" Le$$a :*;*.0 2+a 3 a, / 2+a 3 a,2

we ge% 2+a 3 a,2 / 2+6,*

N)w (re3$4#%i(#"ing 2)%h &i'e& 2" a 3 a

We )2%ain

7+a 3 a,2+a 3 a,28 / +a 3 a,2+6,

An' %hi& gi!e&0 2" i'e$()%en"0 %ha%

+a 3 a,2 / +a3 a,2+6,*

Again0 4&ing 2 / 20 we ge%0

a2 3 a2 / a2+6, – a2+6,<<<**E=n*.

>r)$ E=n*; ? E=n*. we ge%

A2+6, – a2+6, / a2+6, – a2+6,

/9a2+6, / a2+6, 1)r eah a0 20 ϵ N

Hene0 2" 4&ing %he re&4#% @4&% )2%aine'0

We ha!e %ha% a2+6, / aa2+6,

/ aa+6,

/ a.+6,

/ a+6, 1)r eah a02 ϵ N

Again 1r)$ E=n*; we ge%

a2 3 a2 / a2+6,3a2+6,

/ a+6,3a+6,

/ 6

There1)re a2 / a2 1)r eah a020 ϵ N*



43

!eorem:3.1.$* Le% N 'en)%e a 2))#ean near3ring &4h %ha%0 i1 eah )1 -0" ϵ N0

%hen %here e-i&%& an e ϵ N &4h %ha% e- / - an' e" / "* Then N i& a B))#ean

Ring*

Proof: Le% - ϵ N.

C)n&i'er - an' - - *

B" a&&4$(%i)n0 %here e-i&%& an i'e$()%en% e ϵ N &4h %ha%

e- " - an' e+- % -, " & % &.

Th4&0 - - / e - e-

/ +e e,-

/ +e e, . -

/ 7e+ e e, e+e e,8-

/ e+e e,- e+e e, -

/ e+e-e -,e+ e-e- ,

/ e+-- ,e+- -,

/ +--,+--,

Hene0 i% 1)##)w& %ha% -- / 6*

There1)re +N0 , i& an a2e#ian gr)4(*

N)w0 #e% - 0" ϵ N*

Then a)r'ing %) )4r a&&4$(%i)n0 %here e-i&%& an i'e$()%en% 1 ϵ N

S4h %ha% 1 - / - an' 1" / "*

B" The)re$ :*;*:0 -" / + 1- ,"

/ 1-"

/ 1"-

/ +1",-

/ "- *

+i*e*, -"/ "- *

Wi%h %he $4#%i(#ia%i)n 2eing )$$4%a%i!e 0 i% 1)##)w& %ha% N i& a 2))#ean ring*



44

!eorem:3.1.'* E!er" '*g* 2))#ean near3ring N i& a 2))#ean ring*

Proof. Le% N 'en)%e a '*g* 2))#ean near3ring*

S4(()&e S i& a $4#%i(#ia%i!e &e$igr)4( wh)&e e#e$en%& S genera%e +N0 ,

Sa%i&1" & +- " , / &- & " 1)r eah - 0" ϵ N*

I% i& ea&" %) &ee %ha%0 1)r eah &0& ; 0 & . ϵ S an' - ϵ N0 &6 / 60 -6 / 60

& & / 60 an' &; & . / &.&;*

Hene &+- -, / &- &-

/ + & &,-

/ 6-

/ 6*

Ne-%0 1)r - 0" ϵ N0

#e% " / &; &. <<&n 0 where eah & i ϵ S *

Then "+- -, / +&; &. <<&n,+- -,

/ &;+- -,&.+--0,<<<*&n+--,

/ 6*

Th4&0 2" Le$$a :*;*.0

- - / +- -,-

/ +- -,- +- -,

/ +- -,6

/ 6

Tha% i& 0 eah n)n3er) e#e$en% in +N0 , i& )1 )r'er .*

Hene +N0 , i& an a2e#ian gr)4(*

C)n&e=4en%#"0 N i& a ring &ine +N0 , 2eing a2e#ian i$(#ie& 2" an e#e$en%ar"



45

re&4#% %ha% N i& #e1% 'i&%ri24%i!e*

There1)re N i& a 2))#ean ring*

( s)e*ial *lass of Boolean Near-Rings

Le% R 'en)%e a B))#ean ring* Le% eah )1 A an' B 'en)%e a &42ring )1 R

&4h %ha% 3 A B/ 6 an' &4(()&e a2 / 6 1)r eah a ϵ A an' 2 ϵ B *

Ta5e N %) 2e %he &e% )1 a## $a((ing&F

1 F R 9 R &4h %ha%0 1)r eah - ϵ R0 1-, / a- 20 where a ϵ A an' 2 ϵ B*

Then +N0 ⊕0 , i& a B))#ean near3ring where ⊕” an' J'en)%e )r'inar" a''i%i)n an'

)$()&i%i)n )1 $a((ing&0 re&(e%i!e#"*

>ina##"0 +N0 ⊕0 , i& 2))#ean &ine0 1)r eah - ϵ R0

+1*1,+-, / 1+1+-,,

/ a+a-2,2

/ a-a22

/ a-+6,2

/ a-2

/ 1+-,

I% i& %hi& #a&& )1 2))#ean near3ring& whih we wi## hara%erie in %he 1)##)wing

$anner*

Le% A 'en)%e a B))#ean ring an' #e% B 'en)%e an a''i%i!e a2e#ian gr)4(*



46

C)n&i'er %he gr)4( 'ire% &4$ A ⊕ B )1 A an' B*

De1ine a $4#%i(#ia%i)n in A ⊕ B 2"

+a; 02;, +a.02., /+ a;a.02;,

I% an 2e !eri1ie' 'ire%#" %ha% A ⊕ B 1)r$& a 2))#ean righ% near3ring wi%h

)$$4%a%i!e a''i%i)n an' &a%i&1ie& %he i'en%i%" +-3",6 / -"3"-*

We wi## 'en)%e %hi& 2))#ean near3ring 2" N+A0 B,*

!eorem 3.1.+* Le% N 'en)%e a 2))#ean near3ring in whih %he a''i%i)n i&

)$$4%a%i!e* S4(()&e0 1)r eah -0" ϵ N0 %ha% +- 3 ",6 / -" 3 "-*

Then %here e-i&% a 2))#ean ring A an' an a2e#ian gr)4( B &4h %ha% N ≅ N+A0 B,*

Proof: #e% A " ,a ϵ N a " / an' #e% B " ,0 ϵ N26 " 0/.

C#ear#"0 A an' B are a''i%i!e &42gr)4(& )1 N*

>)r eah a;0a. ϵ A0 we ha!e 2" +, %ha%

a;a. –a;a. / +a;3a.,6

/ a;63a.6

/ 636

/ 6

An' %h4& a;a. / a.a;*

A#&)0 A i& #)&e' wi%h re&(e% %) $4#%i(#ia%i)n &ine

+a;a. / a;+a.6,

/ a;6

/ 6 1)r eah a;0a. ϵ A

An' %h4& a;a. ϵ A*

Hene A i& a 2))#ean ring*

>4r%her$)re0 A B / 6 an'0 1r)$ %he 'e1ini%i)n& )1 A an' B a#)ng wi%h

The)re$ :*;*:0

a0 / a0

/ a0

/ a

/ 60 1)r eah a ϵ A an' 0 ϵ B*

Le% + F N N+A0B, 'en)%e a $a((ing 'e1ine' 2"

+-, / +-3 -60-6, 1)r eah - ϵ N*



47

I% i& ea&" %) &ee %ha% i& a''i%i!e*

T) &ee %ha% i& a#&) $4#%i(#ia%i!e0

#e% -;0 -. ϵ N*

U&ing %he i'en%i%" +,0 we )2%ain

-; +-.3-.6, 3 &2-&2&1 " &1-&2-&24

" &1-&2-&2

" &1-&2% &2

" &1

Th4& - ; +- 2 3 -26, – -.-; -.6/ -;6

+i*e*, - ; +- 2 3 -26, / -.-; +-; -.,6

/ -.-; -;-. 3 -.-;

/ -;-.

A#&) 2" %he)re$ :*;*.0

-;-.6 / -;6-.

/ -;6

Th4& +-;, +-.,/+-; – -;60 -;6,+ -. – -.60 -.6,

/++-; – -;6,+ -. – -.6,0-;6,

/+-;+-. – -.6, 3 -;6 + -. – -.6,0 -;6,

/+-;-. 3 -;60 -;6,

/+-;-. 3 -;-.60 -;-.6,

/ +- ; - . ,

Hene i& a H)$)$)r(hi&$ an' i& in@e%i!e i& %ri!ia#*

N)w 1)r eah +a02, N+a02,0 Le% / a 2ϵ

Then 6 / +a2,6

/ a6 26

/ 6 2

/ 2

An' –6 / a 2 – 2



48

/ a

Th4& + , / + – 60 6,

/ +a 02,

Th4& i& S4r @e % i!e *

There1)re i& an i &)$)r(hi&$ an' )n&e=4 en% #" N ≅ N+A0 B,

3.2B55L6(N δ N6(R -R7N89

Remark:3.2.1

We rea## %ha% a near3ring N i& &ai' %) 2e a B))#ean near3ring i1 a.

/ a

1)r a## a in N*

We n)w (r)()&e a $e%h)' %) )n&%r4% B))#ean 3near3ring&*

6&am)le 3.2.2

Le% +G0 , 2e a gr)4( whih )n%ain& a &42gr)4( H an' a n)ner)

n)r$a# &42gr)4( K &4h %ha% %he 1)##)wing )n'i%i)n& are &a%i&1ie' F

+;, H )n%ain& n) n)ner) n)r$a# &42gr)4(& )1 G

+;,G / H K

+.,H ∩ K / 6*

>)r an" a0 2 ∈ G0 'e1ine

a2 / a i1 2 ∉ H

/ h i1 2 ∈ H where h i& gi!en 2" %he 'e)$()&i%i)n )1 a a&0

a / 5 h0 where 5 ∈ K an' h ∈ H *

I% i& ea&" %) !eri1" %ha% +G0 0, i& a near3ring* In %he 1)##)wing we &h)w %ha% i% i& a

B))#ean 3near3ring* An" near3ring )1 1)r$ :*.*. whih ha& a #e1% i'en%i%" i& %he %w)

e#e$en% 1ie#'*

Lemma:3.2.3

The near3ring +G0 0, )n&%r4%e' in E-a$(#e :*.*. i& a B))#ean

3near3ring*



49

Proof:

Le% a ∈ G*

I1 a ∉ H0 %hen #ear#" a. / a*

I1 a ∈ H0 %hen a#&) a. / a 2" %he 'e1ini%i)n )1 $4#%i(#ia%i)n*

Th4& G i& a B))#ean near3ring*

O2!i)4&#" G6 / a ∈ GQa6 / 6 / K

an' G / a ∈ GQa6 / a / H*

Le% a0 2 ∈ G wi%h a2 / 6 an' #e% n ∈ N*

I1 a / 60 %hen an2 / 6*

S4(()&e a ≠ 6 an' a / 5h 1)r &)$e 5 ∈ K an' h ∈ H *

Sine a2 / 60 i% 1)##)w& %ha% 2 $4&% 2e in H*

S) h / a2 / 6*

I1 n ∈ H0 %hen an2 / h2

/ 62

/ 6*

S4(()&e n

∉

H* Then an2 / a2

/ 6*

Th4& in an" a&e0 an2 / 6 an' hene G ha& %he I>P*

S) G6 i& a n)ner) i'ea# )1 G*

We n)w &h)w %ha% G6 i& %he &$a##e&% n)ner) i'ea# )1 G*

>)r0

Le% A 2e a n)ner) i'ea# )1 G*



50

I1 A ∩ G6 / 60

Then A / A ∩ G*

S) A ⊆ G / H

/9/ %) %he h"()%he&i& %ha% H )n%ain& n) n)ner) n)r$a# &42gr)4(& )1 G*

There1)re A ∩ G6 ≠ 6*

Ch))&e 6 ≠ a ∈ A ∩ G6*

N)w a ∉ G / H*

>)r an" - in G60 - / -a

/ -+6 a, – -6 ∈ A*

Th4& G6 / A*

Hene G6 i& %he &$a##e&% n)ner) i'ea# )1 G an' G i& a B))#ean δ near3ring*

S) G i& a B))#ean δ 3near3ring

6&am)le:3.2.$

Le% +N , 2e an" gr)4(*

A&&4$e %ha% %he in%er&e%i)n )1 a## n)ner) n)r$a# &42gr)4(& )1 N i& n)ner)*

De1ine $4#%i(#ia%i)n in N a& 1)##)w& F

a2 / a 1)r a## a an' 2 in N*

Then N i& a &42'ire%#" irre'4i2#e B))#ean near3ring&*

An" near ring )1 1)r$ :*.* i& a n)ner) )n&%an% near3ring an' hene ann)%

()&&e&& a #e1% i'en%i%"*

Lemma:3.2.'

Le% N 2e a n)n)n&%an% B))#ean I>P near3ring* I1 N i& &42'ire%#"

irre'4i2#e0 %hen N )n%ain& n) n)ner) n)r$a# &42gr)4(& )1 N*



51

Proof :

S4(()&e N i& &42'ire%#" irre'4i2#e*

Le% H 2e a n)ner) n)r$a# &42gr)4( )1 N &4h %ha% H ⊆ NC*

Then H ∩ N6 ⊆ N6 ∩ N / 6 an' hene H n N6 / 6*

Sine N ha& I>P an' N i& a n)n)n&%an% near3ring0 N6 i& a n)ner) i'ea# )1 N*

We n)w &h)w %ha% H i& an i'ea# )1 N*

Le% a ∈ H an' n ∈ N*

Sine H ⊆ N 0 an / a ∈ H*

Le% $0 n ∈ N an' a ∈ H*

I1 n ∈ N0 n a ∈ N*

Sine N6 i& an i'ea# )1 N0

N6 N / 6*

Hene $+n a, 3 $n / $6 – $6 / 6 ∈ H*

S4(()&e n ∉ N*

Then n a ∉ N*

Then N i& a δ 3near3ring*

S)0 n an' n a are righ% i'en%i%ie& )1 N*

S) $+n a, 3 $n / $ 3 $ / 6 ∈ H*

Th4& H i& an i'ea# )1 N*

Sine N i& &42'ire%#" irre'4i2#e0 ei%her H / 6 )r N6 / 60 a )n%ra'i%i)n*

Th4& N )n%ain& n) n)ner) n)r$a# &42gr)4(& )1 N*

Corollary:3.2.+

Le% N 2e a n)n3)n&%an% B))#ean I>P near3ring* Then N i& &42'ire%#"

irre'4i2#e i1 an' )n#" i1 N i& genera#ie' in%egra# an' N )n%ain& n) n)ner) n)r$a#

&42gr)4(& )1 N*

Proof :

/9 B" Le$$a :*.*0 N )n%ain& n) n)ner) n)r$a# &42gr)4(& )1 N*

B" The)re$ *.*0 N i& genera#ie' in%egra#*



52

/ F Sine N i& n)n)n&%an% near3ring an' N ha& I>P0 N6 i& a n)ner) i'ea# )1 N*

Le% A 2e a n)ner) i'ea# )1 N*

I1 A ∩ N6 / 60 %hen A ⊆ N 0 whih i& a )n%ra'i%i)n %) %he h"()%he&i&*

S) A ∩ N6 ≠ 6*

Th4& N i& genera#ie' in%egra# &4h %ha% 1)r an" n)ner) i'ea# A0 A ∩ N6 ≠ 6*

There1)re0 N i& &42 'ire%#" irre'4i2#e*

Lemma:3.2.

Le% N 2e a n)n)n&%an% B))#ean I>P near3ring*

I1 N i& &42'ire%#" irre'4i2#e0 %hen N i& a near3ring )1 %he 1)r$ gi!en in

E-a$(#e :*.*.*

Proof:

Sine N i& &42'ire%#" irre'4i2#e0 2" %he a2)!e #e$$a0 N )n%ain& n)

n)ner) n)r$a# &42gr)4(& )1 N*

Ta5e H / N an' K / N6 in E-a$(#e :*.*.

Le% a ∈ N an' a / a6 a 0 where a6 ∈ N60 a ∈ N*

I1 2 ∉ N0 %hen a2 / a*

Le% 2 ∈ N*

B" %he I>P0 N6 i& an i'ea# )1 N*

Hene a6 2 / 6 an' &) a2 / +a6a, 2

/ a6 2 a

/ a*

Thi& )$(#e%e& %he (r))1*



53

C;(P6R-7<

=5R6 CL(9969 5> N6(R-R7N89

$.1 7>P - N6(R-R7N89

Definition:$.1.1

A near ring N i& &ai' %) 14#1i## %he in&er%i)n )1 1a%)r&3

(r)(er%" +I>P, (r)!i'e' %ha% V a 020 n N F + a2 / 6 /9 an2 / 6,*N ha&ϵ

%he &%r)ng I>P i1 e!er" h)$)$)r(hi i$age )1 N ha& %he I>P*

Remark:$.1.2

N ha& %he &%r)ng I>P /9 V I ⊆ N V a0 20 n NFϵ

+a2 I/9 an2 I,*ϵ ϵ

Remark:$.1.3

The 1)##)wing a&&er%i)n& are e=4i!a#en%F

+a, N ha& %he I>P3(r)(er%"*

+2, V n NF +6F n,ϵ ⊆ N*

+, V S ⊆ NF +6 F S, ⊆ N*

O2&er!e %ha% e!er" I>P3near3ring N wi%h #e1% i'en%i%" e i& in 0ȵ

1)r e6 / 6 i$(#ie& %ha% en6 / 60 whene n6 / 6 1)r a## n N*ϵ

Definition$.1.$

C)n&i'er %he 1)##)wing (r)(er%ie&F



54

+P6,F V - ϵ N ∃ n+-,9;F - n +- , / -

+P;,F +P6, an' N - 0"N : n+- 0",9;F +-"3"-,ϵ ȵ n- 0 "3 / -"3"- an'

+P.,F V -0 " ϵ N ∃ n+-0",9;F +-" – "-, n +- 0" , / -" – "- an' N ϵ ȵ

+P:,F V -0 "0 ϵ NF-" / -" +Wea5 C)$$4%a%i!i%",

+P,F V -0 " ϵ N V I ⊆ NF -" I /9 "- Iϵ ϵ

Remarks:$.1.'

+a,The -n +- , / - (r)(er%" ')e& n)% i$(#" %ha% N 1)rϵ ȵ

e!er" N nϵ 14#1i##& i%* Near Ring& wi%h +P ;, are a##e' L3near3rinn &

+2,A2e#ian near ring &X N wi%h V - ϵ NF -.

/ - an' +P :, are – near ring&J*

Pro)osition:$.1.+

+a,+P;, /9 +P., /9 +P,

+2,Eah )ne )1 +P;, %) +P, i$(#ie& %ha% %he &%r)ng I>P – Pr)(er%"

Proof:

+a, +P;, /9 +P., i& i$$e'ia%e*+B" De1ini%i)n,

A&&4$e +P., an' -" Iϵ

Then "- – -" / "-+$)' I,

An' There e-i&%& n IN – ; &4h %ha%ϵ

"-3-" / +"-3-",n

/ +"-, n

/ "-"-** *"-

/ 6 +$)' I,*

Hene "- Iϵ

+2, Sine +P;, 3 +P :, are inheri%e' %) h)$)$)r(hi i$age& i% &411ie& %)

&h)w %he I>P3(r)(er%" in %hi& a&e*

B" +a,0 we )n#" ha!e %) #))5 a% +P :, an' +P,

+P:,F I1 a2 / 6 an' n Nϵ

Then an2 / a2n



55

/ 6n

/ 6*

+P,F I1 a2 # an' n Nϵ ϵ

Then 2a Iϵ

Hene 2+an, Iϵ

There1)re an2 I 2" +Pϵ ,*

Definition:$.1.

A nr* N i& a##e' a#$)& % &$a## ha& a% i1 N $)&% .

'i11eren% annihi#a%)r i'ea#&*

Pro)osition:$.1.?

Le% N / N 6 2e &42'ire%#" irre'4i2#e*

+a,N ha& %he I>P0 24% n) ni#()%en% e#e$en%& 2e&i'e 6 /9N i& in%egra# *

+2,I1 N ha& +P .,0 i& in%egra# an' ha& N ' Z 6

Then N 14#1i##& 2)%h ane##a%i)n #aw&0 i& a2e#ian an'

ei%her )$$4%a%i!e )r ϵ ɳ ;

Proof:

We $a" a&&4$e %ha% N Z 6*

+a, C)n&i'er an" - N*ϵ

The &e$igr)4( +- 5 Q 5 IN 0 ,')e& n)%ϵ

)n%ain 6 an' i& )n%aine' in a $4#%i(#ia%i!e &e$igr)4( M - $a-i$a# 1)r n)% )n%aining 6*

C)n&i'er I -FU

mϵ M x +6F$,*

The I>P i$(#ie& %ha% I - ⊆ N*

Sine - ∉ I- +M- i& # )&e' w*r*%* $4#%i(#i a%i)n,



56

⋂

zϵ N ¿ I / 6

Sine N i& &42'ire%#" irre'4i2#e0 ∃ " NF Iϵ " / 6*

I1 n N i& n)% in Mϵ " 0 %he &42&e$igr)4( 2" M " an' n )n%ain& 6*

S) &)$e (r)'4% )n%aining a%#ea&% )ne %i$e& n +an' ()&&i2#" e#e$en%&

)1 M" , $4&% 2e er)* S4h a (r)'4% ha& )ne )1 %he 1)##)wing 1)r$&F

$; $. / 6 n$[ / 6 $n / 6 n / 6 +$ ; 0 $. 0 $[0$ Mϵ " ,*

An a((#ia%i)n )1+P, "ie#'& n / 6 )r ∃ $ Mϵ "F n$ / 60 in whih a&e

n Iϵ " /6

Again n / 6*

Th4& M" / N an' N i& in%egra#*

+2, +;, I1 ∃ -0 " NF -"3"- Z 60 %a5e 5 IN 3 #ϵ ϵ

wi%h +-"3"-,5 / -"3"-*

+-"3"-,5 3 ;

/ e i& a n)ner) i'e$()%en%*Le% ' Nϵ ' 0 2e Z 6 an' Le% n 2e ar2i%rar" N*ϵ

Then +ne3n,e /60

S) ne / n

An' '+en3n, / 'en3'n

/ 'n3'n

/6

Hene en / n*

S) N ha& an i'en%i%" ; an' eah n)n3er) i'e$()%en% / ;*

+., N)w #e% a0 20 N wi%h a2 / a0 a Z 6*ϵ

I1 a i& en%ra#0 we ge% 2 / *

I1 a i& n)% en%ra#0

∃ 1 NF a131a Z 6ϵ

Hene +a131a,a Z 6



57

Le% l IN 3 # 2e &4h %ha%ϵ

(a( f a)−(f a)a)l

/ a+1a,3+1a,a*

Then (a( f a)−(f a)a)l−1

/ ; 2" +;, an' a ha& a #e1%

in!er&e whih again re&4#%& in 2 / *

+:,We n)w &h)w %ha% +N0, i& a2e#ian*

Le% ;; / .*

I 1 . / 6 0 eah e#e$en% )1 N i& )1 )r'er . an' N i& a2e#ian*

I1 . Z 6 24% . i& en%ra#0 %hen e-(an'ing +n $,+;;, in 2)%h wa"&

gi!e& n $ / $ n 1)r a# # $0 n Nϵ

S) again N i& a2e#ian*

I1 . Z 6 an' . i& n)% en%ra# 02" %he )n&i'era%i)n& a2)!e0

N ϵ ɳ ; an' n+3#, / n

/ n+#,

/9 n / 6*

. ha& a #e1% in!er&e 4 +&a",*

Then 4 i& a righ% in!er&e*

>)r 0

+; 3.4,. / .3.4.

/ .3.*;

/ .3.

/ 6

/9 .4 / ;*

Le% r N 2e ar2i%rar"0 h / 4*r ϵ



58

Then h h / 4r4r

/ +44,r

/ +.4,r

/ ;*r

/ r

>ina##"0 r / h h / h[ h[

/9 .h / . h[

/9 h / h[ *

Th4& +N0, i& a2e#ian*

!eorem:$.1.@

Le% N 2e a 'g near ring*N ha& +P ., /9 N i& )$$4%a%i!e*

Proof:

S4(()&e N ha& +P .,

T) (r)!e Ni& C)$$4%a%i!e

De)$()&e N in%) &42'ire%#" irre'4i2#e near ring&

N i Z 6

C)n&i'er &)$e N i

N i ha& a#&) +P ., an' +N i,' Z 6*

+ a, We 1 ir &% &h)w %ha% eah n i#() %en% e# e$en% i & en %r a# *

We wi## a)$(#i&h %hi& 2 " in'4% i)n )n %he 'egree 5 )1 ni# ()%ene*

5 / ; i& %ri!ia#0 24% we a#&) nee' 5 / .

S4(()&e %ha% n . / 6*

Then V - NF +-n3n-,-n / -n-n3n--nϵ



59

/ 636

/ 6

Sine nn / 6 i $(#ie& n-n / 6 an' n--n / 6 2" %he I>P*

Si$i#ar#"0 +-n3n-,n- / 6

S) +-n3n-,. / 6

Hene -n3n- / 6 2" +P . ,

N)w a&&4$e %ha% V n NF nϵ53 ; / 6 /9 n C+N,0ϵ

An' %a5e $ N wi%h $ϵ

5

/ 6

Then +$53 ;,. / 6

S) $53 ; C+N,ϵ

Hene V - NFϵ 6 / $5 -3-$5

/ $-$5 3 ; –-$$5 3;

/ +$-3-$,$5 3 ;

/ +I>P\,

/ +$-3-$,$+-$3-$,$<<**+$ -3 -$ ,$

/ + +$-3-$,$, 5 3 ;

/ +$+-$,3 +-$,$, 5 3 ;

A((#"ing +P., again "ie#'& $+-$, 3+- $,$ / 6

A& a2)!e0 i% %4rn& )4% %ha% +$-3-$,. / 6*

Hene $-3-$ / 6 *

+2,>r)$ +a, an' %he I>P (r)(er%" 0we ge% %ha% %he &e% N (% +Ni, )1 a## ni#()%en% e#e$en%&

)1 Ni 1)r$& an i'ea#*



60

+,I1 N (% +Ni, / Ni0+P., in&%an%#" re&4#%& %he )$$4%a%i!i%" )1 +N0 ,

+',I1 N (% +N i, / 6 an' N ϵ ɳ ; 0 N i i& in%egra# 2" *;*+a,0 an' hene

a2e#ian 2" *;*+2,*

C)n&e=4en%#" a ring an' %here1)re a )$$4%a%i!e )ne

I1 N ∉ ɳ ; 0 N i& )$$4%a%i!e 2" *;*+2,*

+e, I1 6 Z N (% +Ni, Z Ni0

C)n&i'er´ N i F / Ni Q N (%+N i,

´ N i ha& n) n)n3er) ni#()%en% e#e$en%&0 24% i& again

'g* wi%h +P .,

B" +',´ N i i& a )$$4%a%i!e ring*

S) 1)r a## n[ 0 n N0ϵ

n[n 3 nn[ Nϵ ( %+N i,

Th4& we ge% n[n / nn[ 2" +P .,*

+1,Sine a## N i are )$$4%a%i!e near3ring&0 %he &a$e a((#ie& %) N*

+ g , T h e ) n ! e r & e i & % r i ! i a # *

$ . 2 R 6 8 A L ( R 7 > P N 6 ( R R 7 N 8 9

Definition: $ . 2 . 1

A near ring N i& a##e' a reg4#ar near ring i1 1)r eah a N0 a / a2 aϵ

1)r &)$e 2 Nϵ



61

Lemma: $.2.2

S4(()&e N ha& %he I>P* Then 1)r an" -0 n Nϵ an' e. / e0 -ne / -ene*

Proof F

Le% -0 n Nϵ an' e. / e*

Sine +- 3 -e, e / 60

B" I>P0 we ha!e

+- 3 -e, ne / 6

There1)re -ne / -ene*

Lemma:$.2.3

S4(()&e %ha% N i& a reg4#ar I>P near3ring* Then %he 1)##)wing are %r4eF

+i, a Nϵ a. a. N 1)r a## a ϵ N*

+ii, N ha& n) n)ner) ni#()%en% e#e$en%&*

+iii, N ha& %he &%r)ng I>P*

Proof :

+i,Le% a N*ϵ

Sine N i& reg4#ar0 a / a-a 1)r &)$e - in N*

Sine a- an' -a are i'e$()%en%&0 2" Le$$a *.*.0

-a / -a +-a,

/ - +-a, a +-a,

/ -. a.

An' a- / a- +a-,

/ a+a-, -+a-,

/ a. -. a-*



62

There1)re a / a-a ϵ Na. a. N*

+ii, Sine 2" +i,0 a ϵ Na. a. N 1)r a## a ϵ N

I% 1)##)w& %ha%0 N ha& n) n)n3er) ni#()%en% e#e$en%&

+iii,Le% I 2e an" i'ea# )1 N

An' #e% a0 2 Nϵ wi%h a2 ϵ I*

Sine N i& reg4#ar0

2 / 2"2 1)r &)$e " ϵ N*

Sine 2" i& an i'e$()%en%0 2" Le$$a *.*.0

we ha!e an2 / an+2",2

/ a +2", n+2",2

/ a2"n2 ϵ a2N ⊆ I 1)r a## n ϵ N*

Hene 2" Le$$a *.*. 0

N ha& %he &%r)ng I>P*

Definition:$.2.$

A near3ring N i& &ai' %) 2e #e1% &i$(#e i1 Na / N 1)r a## a ∉ N*

Remark:$.2.'

We wri%e +6F -, / " ϵ NQ"- / 6*

I1 N ha& %he I>P %hen 1)r an" n)ne$(%" &42&e% S )1 N0 +6 F S, /∩

x ϵ S +6F -, i& an

i'ea# )1 N* In (ar%i4#ar0 N6 / +6F6, i& an i'ea# )1 N*

In %hi& a&e i'ea#& )1 %he 1)r$ +6 F S, are a##e' anhi#a%)r i'ea#&*

Definition: $.2.+

A near3ring N i& &ai' %) 2e a genera#ie' in%egra# near3ring i1 +6F -, /

6 1)r a## - ∉ N*



63

Lemma: $.2.

I1 a near3ring N i& reg4#ar an' genera#ie' in%egra# %hen i% i& #e1% &i$(#e*

Proof :

S4(()&e a ∉ N*

Sine N i& reg4#ar0 %here e-i&%& an i'e$()%en% e &4h %ha% Na / Ne*

N)w e ∉ N*

>)r0

I1 e ϵ N0

Sine N i& an in!arian% &42near3ring0

Ne ⊆ N an' &) a ] N

a )n%ra'i%i)n*

B" h"()%he&i&0

+6 e, / 6 an' hene e i& a righ% i'en%i%" )1 N*

Th4& Ne / N an' &) Na / N0

Th4& N i& #e1% &i$(#e*

!eorem: $.2.? Le% N 2e a n)n)n&%an% near3ring* I1 N i& a reg4#ar0 I>P near ring0 %hen

%he 1)##)wing are e=4i!a#en%*

+i, N i& &42'ire%#" irre'4i2#e*

+ii,N i& genera#ie' in%egra# &4h %ha% 1)r an" n)ner) i'ea# I0 I ∩ N ( ≠ 6*

+iii, N6 i& %he &$a##e&% n)ner) i'ea# )1 N*

Proof F

+i, /9 +ii,

Le% A / - ϵ NQ+6F -, Z 6

An' B /∩

xϵ A +6F -,*

B" +i,0 B i& a n)ner) i'ea#*

Sine N i& reg4#ar0 B )n%ain& a n)ner) i'e$()%en% e*

Sine ; ∉ +6 F e,0 e ∉ A an' &) +6 F e, / 6*

N)w i% i& e!i'en% %ha% e i& a righ% i'en%i%" )1 N*



64

Le% - ϵ A*

Then e- / 6 an' hene -e / -6

Sine e i& a righ% i'en%i%"0 - / -e

/ -6 ϵ N*

Th4& A ⊆ N an' N i& genera#ie' in%egra#* The )%her (ar% 1)##)w& 1r)$ %he

)$2ina%i)n )1 +i, an' %he 1a% %ha% N6 i& a n)ner) i'ea#*

+ii, /9 +iii,

Le% A 2e a n)ner) i'ea#*

Then 2" +ii,0 A ∩ N6 Z 6 *

Le% e 2e a n)ner) i'e$()%en% in A ∩ N6*

Sine e ∉ N0 2" +ii,0 + 6 F e , / 6 an' &) e i& a righ% i'en%i%" )1 N*

I1 - ] N60 %hen - / -e

/ - +6 e, – -6 ] A*

Th4& N6 ⊆ A an' N6 i& %he &$a##e&% n)ner) i'ea# )1 N*

+ii,/9 +i,

B" %he)re$ ;*^6 )1 Pi#0 we ha!e

The 1)##)wing )n'i%i)n& 1)r a near ring0 N Z 6 are e=4i!a#en%

+a, N i& &42'ire%#" irre'4i2#e

+2,I1 + I α

¿¿α ∈ A i& a 1a$i#" )1 i'ea#& )1 N wi%h

∩ I α

α ∈ A / 6

%hen ∃α ∈ A : Iα / 6

+,∩I

α

{0}≠ I ⊴ N Z 6

+',N )n%ain& a 4ni=4e $ini$a# i'ea#0 )n%aine' in a## )%her n)n3er) i'ea#&*

N)w0 we ha!e N6 i& %he &$a##e&% n)n3er) i'ea# )1 N*

Th4& N i& &42 'ire%#" irre'4i2#e*



65

Th4& (r)!e'

!eorem:$.2.@

S4(()&e N i& reg4#ar0 I>P an' &423'ire%#" irre'4i2#e* Then 1)r an" -

N0 +6 Fϵ -, / 6 )r +6 F -, / N6* Hene N i& a#$)&% &$a##*

Proof F

I1 N i& a )n&%an% near3ring0

Then #ear#" +6 F -, / 6 1)r a## _ ] N*

A&&4$e %ha% N i& a n)n3)n&%an% near3ring*

Le% - ] N wi%h +6 F -, Z 6 *

B" The)re$ *.*0 N6 ⊆ +6 F -, an' - ϵ N*

Le% a ϵ N*

Then a an 2e 4ni=4e#" wri%%en a& a / a6 a 0 where a6 ϵ N6 0 a ϵ N*

I1 a- / 60 %hen a ϵ N6

Sine 6 / +a6 a, -

/ a6- a

/ a 2ea4&e N6 i& an i'ea# an' N6 N / 6*

Th4& +6 F -, ⊆ N6 an' Hene N6 / +6 F -,*

Hene N i& a#$)&% &$a##*

near rings

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