near rings
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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,
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/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$$a :*;*.0 2+a 3 a, / 2+a 3 a,2
we ge% 2+a 3 a,2 / 2+6,*
N)w (re3$4#%i(#"ing 2)%h &i'e& 2" a 3 a
We )2%ain
7+a 3 a,2+a 3 a,28 / +a 3 a,2+6,
An' %hi& gi!e&0 2" i'e$()%en"0 %ha%
+a 3 a,2 / +a3 a,2+6,*
Again0 4&ing 2 / 20 we ge%0
a2 3 a2 / a2+6, – a2+6,<<<**E=n*.
>r)$ E=n*; ? E=n*. we ge%
A2+6, – a2+6, / a2+6, – a2+6,
/9a2+6, / a2+6, 1)r eah a0 20 ϵ N
Hene0 2" 4&ing %he re&4#% @4&% )2%aine'0
We ha!e %ha% a2+6, / aa2+6,
/ aa+6,
/ a.+6,
/ a+6, 1)r eah a02 ϵ N
Again 1r)$ E=n*; we ge%
a2 3 a2 / a2+6,3a2+6,
/ a+6,3a+6,
/ 6
There1)re a2 / a2 1)r eah a020 ϵ N*
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!eorem:3.1.$* Le% N 'en)%e a 2))#ean near3ring &4h %ha%0 i1 eah )1 -0" ϵ N0
%hen %here e-i&%& an e ϵ N &4h %ha% e- / - an' e" / "* Then N i& a B))#ean
Ring*
Proof: Le% - ϵ N.
C)n&i'er - an' - - *
B" a&&4$(%i)n0 %here e-i&%& an i'e$()%en% e ϵ N &4h %ha%
e- " - an' e+- % -, " & % &.
Th4&0 - - / e - e-
/ +e e,-
/ +e e, . -
/ 7e+ e e, e+e e,8-
/ e+e e,- e+e e, -
/ e+e-e -,e+ e-e- ,
/ e+-- ,e+- -,
/ +--,+--,
Hene0 i% 1)##)w& %ha% -- / 6*
There1)re +N0 , i& an a2e#ian gr)4(*
N)w0 #e% - 0" ϵ N*
Then a)r'ing %) )4r a&&4$(%i)n0 %here e-i&%& an i'e$()%en% 1 ϵ N
S4h %ha% 1 - / - an' 1" / "*
B" The)re$ :*;*:0 -" / + 1- ,"
/ 1-"
/ 1"-
/ +1",-
/ "- *
+i*e*, -"/ "- *
Wi%h %he $4#%i(#ia%i)n 2eing )$$4%a%i!e 0 i% 1)##)w& %ha% N i& a 2))#ean ring*
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!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$$a :*;*.0
- - / +- -,-
/ +- -,- +- -,
/ +- -,6
/ 6
Tha% i& 0 eah n)n3er) e#e$en% in +N0 , i& )1 )r'er .*
Hene +N0 , i& an a2e#ian gr)4(*
C)n&e=4en%#"0 N i& a ring &ine +N0 , 2eing a2e#ian i$(#ie& 2" an e#e$en%ar"
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re&4#% %ha% N i& #e1% 'i&%ri24%i!e*
There1)re N i& a 2))#ean ring*
( s)e*ial *lass of Boolean Near-Rings
Le% R 'en)%e a B))#ean ring* Le% eah )1 A an' B 'en)%e a &42ring )1 R
&4h %ha% 3 A B/ 6 an' &4(()&e a2 / 6 1)r eah a ϵ A an' 2 ϵ B *
Ta5e N %) 2e %he &e% )1 a## $a((ing&F
1 F R 9 R &4h %ha%0 1)r eah - ϵ R0 1-, / a- 20 where a ϵ A an' 2 ϵ B*
Then +N0 ⊕0 , i& a B))#ean near3ring where ⊕” an' J'en)%e )r'inar" a''i%i)n an'
)$()&i%i)n )1 $a((ing&0 re&(e%i!e#"*
>ina##"0 +N0 ⊕0 , i& 2))#ean &ine0 1)r eah - ϵ R0
+1*1,+-, / 1+1+-,,
/ a+a-2,2
/ a-a22
/ a-+6,2
/ a-2
/ 1+-,
I% i& %hi& #a&& )1 2))#ean near3ring& whih we wi## hara%erie in %he 1)##)wing
$anner*
Le% A 'en)%e a B))#ean ring an' #e% B 'en)%e an a''i%i!e a2e#ian gr)4(*
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C)n&i'er %he gr)4( 'ire% &4$ A ⊕ B )1 A an' B*
De1ine a $4#%i(#ia%i)n in A ⊕ B 2"
+a; 02;, +a.02., /+ a;a.02;,
I% an 2e !eri1ie' 'ire%#" %ha% A ⊕ B 1)r$& a 2))#ean righ% near3ring wi%h
)$$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*
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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
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/ 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*
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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*
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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*
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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#*
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/ 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$$a0 N )n%ain& n)
n)ner) n)r$a# &42gr)4(& )1 N*
Ta5e H / N an' K / N6 in E-a$(#e :*.*.
Le% a ∈ N an' a / a6 a 0 where a6 ∈ N60 a ∈ N*
I1 2 ∉ N0 %hen a2 / a*
Le% 2 ∈ N*
B" %he I>P0 N6 i& an i'ea# )1 N*
Hene a6 2 / 6 an' &) a2 / +a6a, 2
/ a6 2 a
/ a*
Thi& )$(#e%e& %he (r))1*
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C;(P6R-7<
=5R6 CL(9969 5> N6(R-R7N89
$.1 7>P - N6(R-R7N89
Definition:$.1.1
A near ring N i& &ai' %) 14#1i## %he in&er%i)n )1 1a%)r&3
(r)(er%" +I>P, (r)!i'e' %ha% V a 020 n N F + a2 / 6 /9 an2 / 6,*N ha&ϵ
%he &%r)ng I>P i1 e!er" h)$)$)r(hi i$age )1 N ha& %he I>P*
Remark:$.1.2
N ha& %he &%r)ng I>P /9 V I ⊆ N V a0 20 n NFϵ
+a2 I/9 an2 I,*ϵ ϵ
Remark:$.1.3
The 1)##)wing a&&er%i)n& are e=4i!a#en%F
+a, N ha& %he I>P3(r)(er%"*
+2, V n NF +6F n,ϵ ⊆ N*
+, V S ⊆ NF +6 F S, ⊆ N*
O2&er!e %ha% e!er" I>P3near3ring N wi%h #e1% i'en%i%" e i& in 0ȵ
1)r e6 / 6 i$(#ie& %ha% en6 / 60 whene n6 / 6 1)r a## n N*ϵ
Definition$.1.$
C)n&i'er %he 1)##)wing (r)(er%ie&F
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+P6,F V - ϵ N ∃ n+-,9;F - n +- , / -
+P;,F +P6, an' N - 0"N : n+- 0",9;F +-"3"-,ϵ ȵ n- 0 "3 / -"3"- an'
+P.,F V -0 " ϵ N ∃ n+-0",9;F +-" – "-, n +- 0" , / -" – "- an' N ϵ ȵ
+P:,F V -0 "0 ϵ NF-" / -" +Wea5 C)$$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$$e'ia%e*+B" De1ini%i)n,
A&&4$e +P., an' -" Iϵ
Then "- – -" / "-+$)' I,
An' There e-i&%& n IN – ; &4h %ha%ϵ
"-3-" / +"-3-",n
/ +"-, n
/ "-"-** *"-
/ 6 +$)' I,*
Hene "- Iϵ
+2, Sine +P;, 3 +P :, are inheri%e' %) h)$)$)r(hi i$age& i% &411ie& %)
&h)w %he I>P3(r)(er%" in %hi& a&e*
B" +a,0 we )n#" ha!e %) #))5 a% +P :, an' +P,
+P:,F I1 a2 / 6 an' n Nϵ
Then an2 / a2n
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/ 6n
/ 6*
+P,F I1 a2 # an' n Nϵ ϵ
Then 2a Iϵ
Hene 2+an, Iϵ
There1)re an2 I 2" +Pϵ ,*
Definition:$.1.
A nr* N i& a##e' a#$)& % &$a## ha& a% i1 N $)&% .
'i11eren% annihi#a%)r i'ea#&*
Pro)osition:$.1.?
Le% N / N 6 2e &42'ire%#" irre'4i2#e*
+a,N ha& %he I>P0 24% n) ni#()%en% e#e$en%& 2e&i'e 6 /9N i& in%egra# *
+2,I1 N ha& +P .,0 i& in%egra# an' ha& N ' Z 6
Then N 14#1i##& 2)%h ane##a%i)n #aw&0 i& a2e#ian an'
ei%her )$$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*%* $4#%i(#i a%i)n,
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⋂
zϵ N ¿ I / 6
Sine N i& &42'ire%#" irre'4i2#e0 ∃ " NF Iϵ " / 6*
I1 n N i& n)% in Mϵ " 0 %he &42&e$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
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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# # $0 n Nϵ
S) again N i& a2e#ian*
I1 . Z 6 an' . i& n)% en%ra# 02" %he )n&i'era%i)n& a2)!e0
N ϵ ɳ ; an' n+3#, / n
/ n+#,
/9 n / 6*
. ha& a #e1% in!er&e 4 +&a",*
Then 4 i& a righ% in!er&e*
>)r 0
+; 3.4,. / .3.4.
/ .3.*;
/ .3.
/ 6
/9 .4 / ;*
Le% r N 2e ar2i%rar"0 h / 4*r ϵ
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Then h h / 4r4r
/ +44,r
/ +.4,r
/ ;*r
/ r
>ina##"0 r / h h / h[ h[
/9 .h / . h[
/9 h / h[ *
Th4& +N0, i& a2e#ian*
!eorem:$.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
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/ 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 ; –-$$5 3;
/ +$-3-$,$5 3 ;
/ +I>P\,
/ +$-3-$,$+-$3-$,$<<**+$ -3 -$ ,$
/ + +$-3-$,$, 5 3 ;
/ +$+-$,3 +-$,$, 5 3 ;
A((#"ing +P., again "ie#'& $+-$, 3+- $,$ / 6
A& a2)!e0 i% %4rn& )4% %ha% +$-3-$,. / 6*
Hene $-3-$ / 6 *
+2,>r)$ +a, an' %he I>P (r)(er%" 0we ge% %ha% %he &e% N (% +Ni, )1 a## ni#()%en% e#e$en%&
)1 Ni 1)r$& an i'ea#*
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+,I1 N (% +Ni, / Ni0+P., in&%an%#" re&4#%& %he )$$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
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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$$a *.*.0
-a / -a +-a,
/ - +-a, a +-a,
/ -. a.
An' a- / a- +a-,
/ a+a-, -+a-,
/ a. -. a-*
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There1)re a / a-a ϵ Na. a. N*
+ii, Sine 2" +i,0 a ϵ Na. a. N 1)r a## a ϵ N
I% 1)##)w& %ha%0 N ha& n) n)n3er) ni#()%en% e#e$en%&
+iii,Le% I 2e an" i'ea# )1 N
An' #e% a0 2 Nϵ wi%h a2 ϵ I*
Sine N i& reg4#ar0
2 / 2"2 1)r &)$e " ϵ N*
Sine 2" i& an i'e$()%en%0 2" Le$$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*
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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$()%en% e*
Sine ; ∉ +6 F e,0 e ∉ A an' &) +6 F e, / 6*
N)w i% i& e!i'en% %ha% e i& a righ% i'en%i%" )1 N*
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Le% - ϵ A*
Then e- / 6 an' hene -e / -6
Sine e i& a righ% i'en%i%"0 - / -e
/ -6 ϵ N*
Th4& A ⊆ N an' N i& genera#ie' in%egra#* The )%her (ar% 1)##)w& 1r)$ %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 &$a##e&% n)n3er) i'ea# )1 N*
Th4& N i& &42 'ire%#" irre'4i2#e*
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Th4& (r)!e'
!eorem:$.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##*
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